9 34

9_33

9_35

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 34's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 34 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X18,11,1,12 X12,17,13,18
Gauss code	1, -4, 3, -7, 6, -1, 2, -3, 5, -6, 8, -9, 7, -5, 4, -2, 9, -8
Dowker-Thistlethwaite code	6 8 10 16 14 18 4 2 12
Conway Notation	[8*20]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{7, 11}, {10, 2}, {11, 4}, {3, 9}, {5, 10}, {4, 8}, {9, 6}, {1, 5}, {2, 7}, {6, 1}, {8, 3}]

[edit Notes on presentations of 9 34]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 34"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X18,11,1,12 X12,17,13,18

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -7, 6, -1, 2, -3, 5, -6, 8, -9, 7, -5, 4, -2, 9, -8

In[6]:=

DTCode[K]

Out[6]=

6 8 10 16 14 18 4 2 12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[8*20]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

$\textrm{BR}(4,\{-1,2,-1,2,-3,2,-1,2,-3\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 4, 9, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{7, 11}, {10, 2}, {11, 4}, {3, 9}, {5, 10}, {4, 8}, {9, 6}, {1, 5}, {2, 7}, {6, 1}, {8, 3}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 14.3446 A-Polynomial See Data:9 34/A-polynomial

[edit Notes for 9 34's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for 9 34's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^3+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3}$
Conway polynomial	$-z^6-z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 69, 0 }
Jones polynomial	$q^4-4 q^3+8 q^2-10 q+12-12 q^{-1} +10 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^6+2 a^2 z^4+z^4 a^{-2} -3 z^4-a^4 z^2+3 a^2 z^2+z^2 a^{-2} -4 z^2+a^2+ a^{-2} -1$
Kauffman polynomial (db, data sources)	$3 a^2 z^8+3 z^8+6 a^3 z^7+14 a z^7+8 z^7 a^{-1} +4 a^4 z^6+5 a^2 z^6+8 z^6 a^{-2} +9 z^6+a^5 z^5-11 a^3 z^5-26 a z^5-10 z^5 a^{-1} +4 z^5 a^{-3} -7 a^4 z^4-19 a^2 z^4-10 z^4 a^{-2} +z^4 a^{-4} -23 z^4-a^5 z^3+5 a^3 z^3+12 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +3 a^4 z^2+10 a^2 z^2+4 z^2 a^{-2} +11 z^2-a z-z a^{-1} -a^2- a^{-2} -1$
The A2 invariant	$-q^{16}+q^{14}+2 q^{12}-2 q^{10}+2 q^8-q^6-q^4+2 q^2-2+3 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12}$
The G2 invariant	$q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+14 q^{72}-12 q^{70}-q^{68}+27 q^{66}-55 q^{64}+83 q^{62}-84 q^{60}+44 q^{58}+24 q^{56}-112 q^{54}+181 q^{52}-188 q^{50}+127 q^{48}-11 q^{46}-114 q^{44}+205 q^{42}-213 q^{40}+135 q^{38}-7 q^{36}-116 q^{34}+167 q^{32}-131 q^{30}+24 q^{28}+103 q^{26}-178 q^{24}+183 q^{22}-102 q^{20}-37 q^{18}+174 q^{16}-269 q^{14}+272 q^{12}-182 q^{10}+32 q^8+130 q^6-244 q^4+280 q^2-217+85 q^{-2} +58 q^{-4} -170 q^{-6} +191 q^{-8} -117 q^{-10} -6 q^{-12} +123 q^{-14} -165 q^{-16} +125 q^{-18} -16 q^{-20} -113 q^{-22} +195 q^{-24} -206 q^{-26} +141 q^{-28} -26 q^{-30} -88 q^{-32} +159 q^{-34} -165 q^{-36} +126 q^{-38} -55 q^{-40} -10 q^{-42} +48 q^{-44} -68 q^{-46} +60 q^{-48} -38 q^{-50} +18 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_34.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+3 q^9-3 q^7+3 q^5-2 q^3+2 q^{-1} -2 q^{-3} +4 q^{-5} -3 q^{-7} + q^{-9}$
2	$q^{32}-3 q^{30}-q^{28}+12 q^{26}-8 q^{24}-16 q^{22}+25 q^{20}+q^{18}-32 q^{16}+22 q^{14}+16 q^{12}-28 q^{10}+6 q^8+19 q^6-10 q^4-13 q^2+11+14 q^{-2} -25 q^{-4} -2 q^{-6} +33 q^{-8} -21 q^{-10} -15 q^{-12} +30 q^{-14} -7 q^{-16} -15 q^{-18} +10 q^{-20} + q^{-22} -3 q^{-24} + q^{-26}$
3	$-q^{63}+3 q^{61}+q^{59}-8 q^{57}-7 q^{55}+16 q^{53}+30 q^{51}-24 q^{49}-64 q^{47}+6 q^{45}+107 q^{43}+40 q^{41}-138 q^{39}-106 q^{37}+135 q^{35}+178 q^{33}-96 q^{31}-232 q^{29}+37 q^{27}+250 q^{25}+28 q^{23}-234 q^{21}-86 q^{19}+198 q^{17}+119 q^{15}-144 q^{13}-137 q^{11}+85 q^9+150 q^7-25 q^5-151 q^3-42 q+150 q^{-1} +110 q^{-3} -132 q^{-5} -180 q^{-7} +98 q^{-9} +234 q^{-11} -44 q^{-13} -255 q^{-15} -22 q^{-17} +239 q^{-19} +81 q^{-21} -187 q^{-23} -113 q^{-25} +121 q^{-27} +105 q^{-29} -53 q^{-31} -82 q^{-33} +15 q^{-35} +48 q^{-37} -2 q^{-39} -18 q^{-41} -3 q^{-43} +7 q^{-45} + q^{-47} -3 q^{-49} + q^{-51}$
4	$q^{104}-3 q^{102}-q^{100}+8 q^{98}+3 q^{96}-q^{94}-30 q^{92}-20 q^{90}+45 q^{88}+67 q^{86}+55 q^{84}-123 q^{82}-211 q^{80}-29 q^{78}+231 q^{76}+437 q^{74}+26 q^{72}-539 q^{70}-606 q^{68}-39 q^{66}+965 q^{64}+866 q^{62}-237 q^{60}-1287 q^{58}-1138 q^{56}+669 q^{54}+1720 q^{52}+979 q^{50}-1012 q^{48}-2141 q^{46}-546 q^{44}+1536 q^{42}+2020 q^{40}+111 q^{38}-2061 q^{36}-1539 q^{34}+565 q^{32}+2042 q^{30}+996 q^{28}-1229 q^{26}-1668 q^{24}-264 q^{22}+1418 q^{20}+1232 q^{18}-399 q^{16}-1348 q^{14}-721 q^{12}+758 q^{10}+1234 q^8+313 q^6-1013 q^4-1169 q^2+46+1278 q^{-2} +1203 q^{-4} -513 q^{-6} -1679 q^{-8} -981 q^{-10} +1003 q^{-12} +2097 q^{-14} +482 q^{-16} -1631 q^{-18} -2013 q^{-20} +11 q^{-22} +2191 q^{-24} +1535 q^{-26} -628 q^{-28} -2103 q^{-30} -1083 q^{-32} +1160 q^{-34} +1627 q^{-36} +495 q^{-38} -1100 q^{-40} -1210 q^{-42} +49 q^{-44} +779 q^{-46} +690 q^{-48} -151 q^{-50} -563 q^{-52} -227 q^{-54} +109 q^{-56} +291 q^{-58} +71 q^{-60} -111 q^{-62} -74 q^{-64} -24 q^{-66} +53 q^{-68} +22 q^{-70} -13 q^{-72} -6 q^{-74} -6 q^{-76} +7 q^{-78} + q^{-80} -3 q^{-82} + q^{-84}$
5	$-q^{155}+3 q^{153}+q^{151}-8 q^{149}-3 q^{147}+5 q^{145}+15 q^{143}+20 q^{141}-q^{139}-59 q^{137}-86 q^{135}-15 q^{133}+123 q^{131}+240 q^{129}+177 q^{127}-129 q^{125}-545 q^{123}-631 q^{121}-100 q^{119}+802 q^{117}+1419 q^{115}+982 q^{113}-623 q^{111}-2384 q^{109}-2611 q^{107}-542 q^{105}+2761 q^{103}+4754 q^{101}+3135 q^{99}-1778 q^{97}-6538 q^{95}-6794 q^{93}-1213 q^{91}+6706 q^{89}+10559 q^{87}+6079 q^{85}-4446 q^{83}-12971 q^{81}-11708 q^{79}-320 q^{77}+12798 q^{75}+16578 q^{73}+6682 q^{71}-9776 q^{69}-19250 q^{67}-13013 q^{65}+4536 q^{63}+19015 q^{61}+17848 q^{59}+1556 q^{57}-16212 q^{55}-20243 q^{53}-7017 q^{51}+11839 q^{49}+20042 q^{47}+10881 q^{45}-7083 q^{43}-17954 q^{41}-12766 q^{39}+2989 q^{37}+14851 q^{35}+12931 q^{33}+56 q^{31}-11650 q^{29}-12114 q^{27}-1995 q^{25}+8933 q^{23}+11006 q^{21}+3244 q^{19}-6807 q^{17}-10249 q^{15}-4442 q^{13}+5184 q^{11}+10107 q^9+6046 q^7-3522 q^5-10451 q^3-8507 q+1299 q^{-1} +10908 q^{-3} +11682 q^{-5} +1980 q^{-7} -10674 q^{-9} -15162 q^{-11} -6449 q^{-13} +9048 q^{-15} +18039 q^{-17} +11723 q^{-19} -5574 q^{-21} -19230 q^{-23} -16847 q^{-25} +368 q^{-27} +17919 q^{-29} +20551 q^{-31} +5661 q^{-33} -13978 q^{-35} -21629 q^{-37} -11107 q^{-39} +8117 q^{-41} +19635 q^{-43} +14545 q^{-45} -1807 q^{-47} -15140 q^{-49} -15083 q^{-51} -3294 q^{-53} +9385 q^{-55} +12979 q^{-57} +6171 q^{-59} -4122 q^{-61} -9312 q^{-63} -6523 q^{-65} +394 q^{-67} +5411 q^{-69} +5270 q^{-71} +1385 q^{-73} -2442 q^{-75} -3377 q^{-77} -1635 q^{-79} +677 q^{-81} +1733 q^{-83} +1211 q^{-85} +37 q^{-87} -739 q^{-89} -650 q^{-91} -151 q^{-93} +224 q^{-95} +283 q^{-97} +113 q^{-99} -60 q^{-101} -110 q^{-103} -41 q^{-105} +20 q^{-107} +27 q^{-109} +11 q^{-111} - q^{-113} -9 q^{-115} -6 q^{-117} +7 q^{-119} + q^{-121} -3 q^{-123} + q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+2 q^{12}-2 q^{10}+2 q^8-q^6-q^4+2 q^2-2+3 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12}$
1,1	$q^{44}-6 q^{42}+20 q^{40}-50 q^{38}+107 q^{36}-204 q^{34}+344 q^{32}-514 q^{30}+701 q^{28}-864 q^{26}+950 q^{24}-938 q^{22}+791 q^{20}-502 q^{18}+98 q^{16}+374 q^{14}-840 q^{12}+1276 q^{10}-1606 q^8+1788 q^6-1806 q^4+1650 q^2-1352+934 q^{-2} -456 q^{-4} -16 q^{-6} +432 q^{-8} -726 q^{-10} +888 q^{-12} -918 q^{-14} +850 q^{-16} -712 q^{-18} +538 q^{-20} -376 q^{-22} +248 q^{-24} -148 q^{-26} +77 q^{-28} -38 q^{-30} +18 q^{-32} -6 q^{-34} + q^{-36}$
2,0	$q^{42}-q^{40}-3 q^{38}+7 q^{34}+5 q^{32}-10 q^{30}-7 q^{28}+9 q^{26}+7 q^{24}-13 q^{22}-7 q^{20}+13 q^{18}+8 q^{16}-11 q^{14}-q^{12}+13 q^{10}-5 q^8-3 q^6+4 q^4-q^2-7+5 q^{-2} +7 q^{-4} -13 q^{-6} -4 q^{-8} +16 q^{-10} +6 q^{-12} -17 q^{-14} + q^{-16} +14 q^{-18} -9 q^{-22} -3 q^{-24} +6 q^{-26} -2 q^{-30} + q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-3 q^{32}+q^{30}+6 q^{28}-12 q^{26}+6 q^{24}+13 q^{22}-22 q^{20}+10 q^{18}+15 q^{16}-24 q^{14}+7 q^{12}+14 q^{10}-13 q^8-2 q^6+7 q^4+3 q^2-6-6 q^{-2} +18 q^{-4} -6 q^{-6} -17 q^{-8} +25 q^{-10} -5 q^{-12} -18 q^{-14} +20 q^{-16} -3 q^{-18} -10 q^{-20} +9 q^{-22} -3 q^{-26} + q^{-28}$
1,0,0	$-q^{21}+q^{19}+2 q^{15}-2 q^{13}+3 q^{11}-2 q^9+q^7-q^5+q^3- q^{-1} +2 q^{-3} -2 q^{-5} +2 q^{-7} - q^{-9} +3 q^{-11} -2 q^{-13} + q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}-q^{42}-3 q^{40}+2 q^{38}+4 q^{36}-4 q^{34}-5 q^{32}+9 q^{30}+7 q^{28}-14 q^{26}-4 q^{24}+20 q^{22}-q^{20}-20 q^{18}+9 q^{16}+17 q^{14}-14 q^{12}-12 q^{10}+16 q^8+2 q^6-19 q^4+10 q^2+18-16 q^{-2} -5 q^{-4} +24 q^{-6} -5 q^{-8} -21 q^{-10} +8 q^{-12} +15 q^{-14} -12 q^{-16} -10 q^{-18} +14 q^{-20} +7 q^{-22} -10 q^{-24} - q^{-26} +7 q^{-28} -2 q^{-30} -2 q^{-32} + q^{-34}$
1,0,0,0	$-q^{26}+q^{24}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+q^8-q^6+q^4-q^2+1-2 q^{-2} +2 q^{-4} -2 q^{-6} +2 q^{-8} +3 q^{-14} -2 q^{-16} + q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+3 q^{32}-7 q^{30}+12 q^{28}-18 q^{26}+24 q^{24}-25 q^{22}+26 q^{20}-22 q^{18}+15 q^{16}-4 q^{14}-9 q^{12}+22 q^{10}-35 q^8+44 q^6-49 q^4+49 q^2-44+34 q^{-2} -20 q^{-4} +8 q^{-6} +5 q^{-8} -15 q^{-10} +23 q^{-12} -26 q^{-14} +26 q^{-16} -23 q^{-18} +18 q^{-20} -11 q^{-22} +6 q^{-24} -3 q^{-26} + q^{-28}$
1,0	$q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+9 q^{46}-q^{44}-15 q^{42}-8 q^{40}+18 q^{38}+19 q^{36}-10 q^{34}-27 q^{32}-3 q^{30}+28 q^{28}+15 q^{26}-21 q^{24}-22 q^{22}+10 q^{20}+23 q^{18}-q^{16}-20 q^{14}-3 q^{12}+18 q^{10}+7 q^8-16 q^6-11 q^4+13 q^2+16-10 q^{-2} -20 q^{-4} +7 q^{-6} +23 q^{-8} + q^{-10} -25 q^{-12} -10 q^{-14} +23 q^{-16} +22 q^{-18} -14 q^{-20} -27 q^{-22} + q^{-24} +23 q^{-26} +10 q^{-28} -13 q^{-30} -14 q^{-32} +3 q^{-34} +10 q^{-36} +3 q^{-38} -3 q^{-40} -3 q^{-42} + q^{-46}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{46}-3 q^{44}+4 q^{42}-6 q^{40}+10 q^{38}-15 q^{36}+17 q^{34}-18 q^{32}+22 q^{30}-21 q^{28}+18 q^{26}-15 q^{24}+14 q^{22}-5 q^{20}-5 q^{18}+10 q^{16}-14 q^{14}+25 q^{12}-33 q^{10}+33 q^8-35 q^6+40 q^4-37 q^2+31-29 q^{-2} +25 q^{-4} -12 q^{-6} +4 q^{-8} -2 q^{-10} -5 q^{-12} +16 q^{-14} -17 q^{-16} +18 q^{-18} -22 q^{-20} +23 q^{-22} -16 q^{-24} +14 q^{-26} -14 q^{-28} +10 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} + q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+14 q^{72}-12 q^{70}-q^{68}+27 q^{66}-55 q^{64}+83 q^{62}-84 q^{60}+44 q^{58}+24 q^{56}-112 q^{54}+181 q^{52}-188 q^{50}+127 q^{48}-11 q^{46}-114 q^{44}+205 q^{42}-213 q^{40}+135 q^{38}-7 q^{36}-116 q^{34}+167 q^{32}-131 q^{30}+24 q^{28}+103 q^{26}-178 q^{24}+183 q^{22}-102 q^{20}-37 q^{18}+174 q^{16}-269 q^{14}+272 q^{12}-182 q^{10}+32 q^8+130 q^6-244 q^4+280 q^2-217+85 q^{-2} +58 q^{-4} -170 q^{-6} +191 q^{-8} -117 q^{-10} -6 q^{-12} +123 q^{-14} -165 q^{-16} +125 q^{-18} -16 q^{-20} -113 q^{-22} +195 q^{-24} -206 q^{-26} +141 q^{-28} -26 q^{-30} -88 q^{-32} +159 q^{-34} -165 q^{-36} +126 q^{-38} -55 q^{-40} -10 q^{-42} +48 q^{-44} -68 q^{-46} +60 q^{-48} -38 q^{-50} +18 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66}$

. </div></div>

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 34"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^3+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^6-z^2+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 69, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^4-4 q^3+8 q^2-10 q+12-12 q^{-1} +10 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^6+2 a^2 z^4+z^4 a^{-2} -3 z^4-a^4 z^2+3 a^2 z^2+z^2 a^{-2} -4 z^2+a^2+ a^{-2} -1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$3 a^2 z^8+3 z^8+6 a^3 z^7+14 a z^7+8 z^7 a^{-1} +4 a^4 z^6+5 a^2 z^6+8 z^6 a^{-2} +9 z^6+a^5 z^5-11 a^3 z^5-26 a z^5-10 z^5 a^{-1} +4 z^5 a^{-3} -7 a^4 z^4-19 a^2 z^4-10 z^4 a^{-2} +z^4 a^{-4} -23 z^4-a^5 z^3+5 a^3 z^3+12 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +3 a^4 z^2+10 a^2 z^2+4 z^2 a^{-2} +11 z^2-a z-z a^{-1} -a^2- a^{-2} -1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n32, K11n119,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 34"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-t^3+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3}$, $q^4-4 q^3+8 q^2-10 q+12-12 q^{-1} +10 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n32, K11n119,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $0$ $8$ $\frac{34}{3}$ $\frac{14}{3}$ $0$ $0$ $32$ $-32$ $-\frac{32}{3}$ $0$ $-\frac{136}{3}$ $-\frac{56}{3}$ $-\frac{1231}{30}$ $-\frac{338}{15}$ $-\frac{1742}{45}$ $\frac{655}{18}$ $-\frac{271}{30}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 9 34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               5 1   4 3             5 3     -2 1           7 5       2 -1         6 6         0 -3       4 6           -2 -5     3 6             3 -7   1 4               -3 -9   3                 3 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

<p>

$n$	$J_n$
2	$q^{12}-4 q^{11}+4 q^{10}+10 q^9-29 q^8+12 q^7+47 q^6-74 q^5+6 q^4+101 q^3-109 q^2-17 q+140-112 q^{-1} -41 q^{-2} +143 q^{-3} -83 q^{-4} -54 q^{-5} +109 q^{-6} -39 q^{-7} -48 q^{-8} +55 q^{-9} -6 q^{-10} -24 q^{-11} +14 q^{-12} +2 q^{-13} -4 q^{-14} + q^{-15}$
3	$q^{24}-4 q^{23}+4 q^{22}+6 q^{21}-9 q^{20}-19 q^{19}+20 q^{18}+56 q^{17}-42 q^{16}-116 q^{15}+49 q^{14}+214 q^{13}-26 q^{12}-350 q^{11}-25 q^{10}+482 q^9+132 q^8-611 q^7-258 q^6+693 q^5+410 q^4-747 q^3-536 q^2+741 q+652-707 q^{-1} -728 q^{-2} +632 q^{-3} +778 q^{-4} -532 q^{-5} -793 q^{-6} +410 q^{-7} +771 q^{-8} -269 q^{-9} -714 q^{-10} +126 q^{-11} +623 q^{-12} -7 q^{-13} -492 q^{-14} -87 q^{-15} +354 q^{-16} +129 q^{-17} -218 q^{-18} -130 q^{-19} +113 q^{-20} +97 q^{-21} -40 q^{-22} -63 q^{-23} +12 q^{-24} +27 q^{-25} -9 q^{-27} -2 q^{-28} +4 q^{-29} - q^{-30}$
4	$q^{40}-4 q^{39}+4 q^{38}+6 q^{37}-13 q^{36}+q^{35}-11 q^{34}+39 q^{33}+37 q^{32}-90 q^{31}-49 q^{30}-48 q^{29}+221 q^{28}+257 q^{27}-272 q^{26}-385 q^{25}-384 q^{24}+633 q^{23}+1098 q^{22}-183 q^{21}-1115 q^{20}-1643 q^{19}+743 q^{18}+2693 q^{17}+949 q^{16}-1582 q^{15}-3886 q^{14}-277 q^{13}+4168 q^{12}+3112 q^{11}-926 q^{10}-6066 q^9-2301 q^8+4550 q^7+5225 q^6+689 q^5-7160 q^4-4285 q^3+3852 q^2+6391 q+2405-7085 q^{-1} -5517 q^{-2} +2637 q^{-3} +6547 q^{-4} +3731 q^{-5} -6164 q^{-6} -5993 q^{-7} +1158 q^{-8} +5920 q^{-9} +4680 q^{-10} -4533 q^{-11} -5807 q^{-12} -524 q^{-13} +4516 q^{-14} +5119 q^{-15} -2308 q^{-16} -4761 q^{-17} -2001 q^{-18} +2412 q^{-19} +4597 q^{-20} -136 q^{-21} -2852 q^{-22} -2485 q^{-23} +330 q^{-24} +3002 q^{-25} +993 q^{-26} -861 q^{-27} -1744 q^{-28} -721 q^{-29} +1195 q^{-30} +844 q^{-31} +189 q^{-32} -641 q^{-33} -622 q^{-34} +191 q^{-35} +277 q^{-36} +256 q^{-37} -76 q^{-38} -211 q^{-39} -15 q^{-40} +17 q^{-41} +74 q^{-42} +12 q^{-43} -33 q^{-44} -3 q^{-45} -5 q^{-46} +9 q^{-47} +2 q^{-48} -4 q^{-49} + q^{-50}$
5	$q^{60}-4 q^{59}+4 q^{58}+6 q^{57}-13 q^{56}-3 q^{55}+9 q^{54}+8 q^{53}+20 q^{52}-q^{51}-74 q^{50}-72 q^{49}+59 q^{48}+181 q^{47}+190 q^{46}-60 q^{45}-449 q^{44}-571 q^{43}-30 q^{42}+957 q^{41}+1364 q^{40}+462 q^{39}-1505 q^{38}-2883 q^{37}-1772 q^{36}+1892 q^{35}+5191 q^{34}+4347 q^{33}-1364 q^{32}-7900 q^{31}-8689 q^{30}-897 q^{29}+10381 q^{28}+14640 q^{27}+5444 q^{26}-11494 q^{25}-21368 q^{24}-12686 q^{23}+10324 q^{22}+27973 q^{21}+21796 q^{20}-6404 q^{19}-32886 q^{18}-31910 q^{17}-198 q^{16}+35624 q^{15}+41435 q^{14}+8486 q^{13}-35518 q^{12}-49461 q^{11}-17413 q^{10}+33241 q^9+55091 q^8+25783 q^7-29202 q^6-58452 q^5-32910 q^4+24528 q^3+59579 q^2+38437 q-19500-59226 q^{-1} -42519 q^{-2} +14722 q^{-3} +57635 q^{-4} +45366 q^{-5} -9932 q^{-6} -55165 q^{-7} -47442 q^{-8} +5096 q^{-9} +51828 q^{-10} +48808 q^{-11} +119 q^{-12} -47403 q^{-13} -49515 q^{-14} -5832 q^{-15} +41709 q^{-16} +49272 q^{-17} +11825 q^{-18} -34528 q^{-19} -47595 q^{-20} -17694 q^{-21} +25954 q^{-22} +44084 q^{-23} +22696 q^{-24} -16564 q^{-25} -38434 q^{-26} -25897 q^{-27} +7098 q^{-28} +30858 q^{-29} +26727 q^{-30} +1204 q^{-31} -22142 q^{-32} -24730 q^{-33} -7381 q^{-34} +13309 q^{-35} +20490 q^{-36} +10678 q^{-37} -5684 q^{-38} -14834 q^{-39} -11161 q^{-40} +191 q^{-41} +9102 q^{-42} +9415 q^{-43} +2841 q^{-44} -4309 q^{-45} -6681 q^{-46} -3662 q^{-47} +1183 q^{-48} +3834 q^{-49} +3097 q^{-50} +451 q^{-51} -1768 q^{-52} -2043 q^{-53} -810 q^{-54} +531 q^{-55} +1028 q^{-56} +678 q^{-57} -7 q^{-58} -438 q^{-59} -373 q^{-60} -86 q^{-61} +126 q^{-62} +147 q^{-63} +79 q^{-64} -22 q^{-65} -67 q^{-66} -23 q^{-67} +9 q^{-68} +9 q^{-69} +8 q^{-70} +5 q^{-71} -9 q^{-72} -2 q^{-73} +4 q^{-74} - q^{-75}$
6	$q^{84}-4 q^{83}+4 q^{82}+6 q^{81}-13 q^{80}-3 q^{79}+5 q^{78}+28 q^{77}-11 q^{76}-18 q^{75}+15 q^{74}-87 q^{73}-16 q^{72}+93 q^{71}+214 q^{70}+60 q^{69}-172 q^{68}-226 q^{67}-596 q^{66}-201 q^{65}+616 q^{64}+1477 q^{63}+1095 q^{62}-260 q^{61}-1676 q^{60}-3767 q^{59}-2705 q^{58}+1083 q^{57}+6329 q^{56}+7879 q^{55}+4218 q^{54}-3277 q^{53}-14376 q^{52}-16750 q^{51}-7389 q^{50}+12518 q^{49}+28184 q^{48}+29057 q^{47}+10529 q^{46}-27268 q^{45}-53389 q^{44}-48973 q^{43}-4566 q^{42}+51710 q^{41}+87565 q^{40}+72590 q^{39}-6610 q^{38}-94578 q^{37}-136646 q^{36}-85215 q^{35}+30912 q^{34}+151409 q^{33}+191466 q^{32}+92051 q^{31}-82084 q^{30}-229420 q^{29}-229647 q^{28}-78266 q^{27}+154170 q^{26}+312752 q^{25}+257079 q^{24}+23502 q^{23}-255646 q^{22}-371665 q^{21}-251033 q^{20}+63988 q^{19}+363413 q^{18}+413536 q^{17}+186123 q^{16}-191750 q^{15}-441171 q^{14}-408894 q^{13}-75882 q^{12}+328123 q^{11}+497760 q^{10}+330085 q^9-83203 q^8-430541 q^7-498059 q^6-197997 q^5+250860 q^4+508808 q^3+414062 q^2+14014 q-380135-523707 q^{-1} -273997 q^{-2} +175604 q^{-3} +482484 q^{-4} +448797 q^{-5} +83170 q^{-6} -323134 q^{-7} -517342 q^{-8} -318813 q^{-9} +109774 q^{-10} +442844 q^{-11} +462028 q^{-12} +141818 q^{-13} -259898 q^{-14} -495577 q^{-15} -356319 q^{-16} +34442 q^{-17} +384479 q^{-18} +463670 q^{-19} +209578 q^{-20} -170599 q^{-21} -447370 q^{-22} -389077 q^{-23} -65372 q^{-24} +286036 q^{-25} +435588 q^{-26} +280278 q^{-27} -44606 q^{-28} -348258 q^{-29} -389585 q^{-30} -173593 q^{-31} +140530 q^{-32} +348024 q^{-33} +315023 q^{-34} +92020 q^{-35} -193995 q^{-36} -321541 q^{-37} -239573 q^{-38} -16250 q^{-39} +199049 q^{-40} +271427 q^{-41} +178557 q^{-42} -29574 q^{-43} -186479 q^{-44} -217687 q^{-45} -115811 q^{-46} +42172 q^{-47} +156660 q^{-48} +170542 q^{-49} +71966 q^{-50} -45861 q^{-51} -123612 q^{-52} -118831 q^{-53} -48992 q^{-54} +38852 q^{-55} +94087 q^{-56} +80016 q^{-57} +30049 q^{-58} -29789 q^{-59} -61282 q^{-60} -55241 q^{-61} -18946 q^{-62} +22114 q^{-63} +38216 q^{-64} +33982 q^{-65} +11198 q^{-66} -11458 q^{-67} -24328 q^{-68} -20110 q^{-69} -5535 q^{-70} +5890 q^{-71} +13020 q^{-72} +10756 q^{-73} +4175 q^{-74} -3742 q^{-75} -6640 q^{-76} -4888 q^{-77} -2203 q^{-78} +1492 q^{-79} +2913 q^{-80} +2695 q^{-81} +623 q^{-82} -680 q^{-83} -1006 q^{-84} -1063 q^{-85} -320 q^{-86} +212 q^{-87} +552 q^{-88} +242 q^{-89} +59 q^{-90} -27 q^{-91} -163 q^{-92} -92 q^{-93} -26 q^{-94} +72 q^{-95} +16 q^{-96} +2 q^{-97} +15 q^{-98} -14 q^{-99} -8 q^{-100} -5 q^{-101} +9 q^{-102} +2 q^{-103} -4 q^{-104} + q^{-105}$
7	$q^{112}-4 q^{111}+4 q^{110}+6 q^{109}-13 q^{108}-3 q^{107}+5 q^{106}+24 q^{105}+9 q^{104}-49 q^{103}-2 q^{102}+2 q^{101}-31 q^{100}+28 q^{99}+75 q^{98}+158 q^{97}+48 q^{96}-291 q^{95}-318 q^{94}-260 q^{93}-66 q^{92}+534 q^{91}+917 q^{90}+1208 q^{89}+482 q^{88}-1471 q^{87}-2844 q^{86}-3329 q^{85}-1711 q^{84}+2371 q^{83}+6470 q^{82}+9138 q^{81}+6836 q^{80}-2254 q^{79}-13194 q^{78}-21631 q^{77}-19779 q^{76}-3481 q^{75}+20489 q^{74}+43605 q^{73}+49176 q^{72}+25233 q^{71}-21814 q^{70}-75414 q^{69}-103193 q^{68}-77368 q^{67}+217 q^{66}+106234 q^{65}+185323 q^{64}+179387 q^{63}+71464 q^{62}-113375 q^{61}-286642 q^{60}-341928 q^{59}-223202 q^{58}+56051 q^{57}+375169 q^{56}+559280 q^{55}+480840 q^{54}+112560 q^{53}-400811 q^{52}-797484 q^{51}-843686 q^{50}-430400 q^{49}+298839 q^{48}+993820 q^{47}+1278907 q^{46}+910274 q^{45}-15765 q^{44}-1073367 q^{43}-1718343 q^{42}-1520174 q^{41}-470669 q^{40}+964613 q^{39}+2073346 q^{38}+2192067 q^{37}+1139954 q^{36}-633602 q^{35}-2263880 q^{34}-2829249 q^{33}-1922465 q^{32}+89307 q^{31}+2235800 q^{30}+3341119 q^{29}+2725638 q^{28}+609765 q^{27}-1986923 q^{26}-3661296 q^{25}-3449927 q^{24}-1377286 q^{23}+1555703 q^{22}+3768099 q^{21}+4023424 q^{20}+2119469 q^{19}-1015128 q^{18}-3681754 q^{17}-4408420 q^{16}-2762212 q^{15}+444233 q^{14}+3454987 q^{13}+4609291 q^{12}+3261430 q^{11}+87493 q^{10}-3149968 q^9-4658362 q^8-3610169 q^7-537899 q^6+2826220 q^5+4604709 q^4+3825894 q^3+890834 q^2-2522244 q-4494135-3945409 q^{-1} -1157376 q^{-2} +2257688 q^{-3} +4364984 q^{-4} +4006145 q^{-5} +1360966 q^{-6} -2029292 q^{-7} -4235883 q^{-8} -4042274 q^{-9} -1535296 q^{-10} +1820136 q^{-11} +4112104 q^{-12} +4076227 q^{-13} +1709872 q^{-14} -1604177 q^{-15} -3981713 q^{-16} -4116003 q^{-17} -1909743 q^{-18} +1351280 q^{-19} +3823412 q^{-20} +4156866 q^{-21} +2147420 q^{-22} -1036621 q^{-23} -3607912 q^{-24} -4177817 q^{-25} -2420560 q^{-26} +641548 q^{-27} +3304202 q^{-28} +4148097 q^{-29} +2710187 q^{-30} -164342 q^{-31} -2887912 q^{-32} -4027779 q^{-33} -2978499 q^{-34} -377124 q^{-35} +2346613 q^{-36} +3778094 q^{-37} +3175544 q^{-38} +942327 q^{-39} -1691930 q^{-40} -3373205 q^{-41} -3244729 q^{-42} -1469069 q^{-43} +962632 q^{-44} +2808633 q^{-45} +3139347 q^{-46} +1886766 q^{-47} -224497 q^{-48} -2115194 q^{-49} -2838187 q^{-50} -2127516 q^{-51} -438709 q^{-52} +1355500 q^{-53} +2354958 q^{-54} +2150543 q^{-55} +946552 q^{-56} -617661 q^{-57} -1746377 q^{-58} -1954553 q^{-59} -1238823 q^{-60} -6866 q^{-61} +1097392 q^{-62} +1583164 q^{-63} +1299853 q^{-64} +447271 q^{-65} -504567 q^{-66} -1116612 q^{-67} -1160225 q^{-68} -672613 q^{-69} +46196 q^{-70} +648950 q^{-71} +889664 q^{-72} +700205 q^{-73} +237065 q^{-74} -260227 q^{-75} -574945 q^{-76} -587886 q^{-77} -349646 q^{-78} -1773 q^{-79} +292569 q^{-80} +407982 q^{-81} +333928 q^{-82} +133949 q^{-83} -89221 q^{-84} -229710 q^{-85} -249493 q^{-86} -162516 q^{-87} -23369 q^{-88} +93644 q^{-89} +149147 q^{-90} +132781 q^{-91} +63605 q^{-92} -14057 q^{-93} -69180 q^{-94} -83861 q^{-95} -59274 q^{-96} -18707 q^{-97} +19848 q^{-98} +41277 q^{-99} +39464 q^{-100} +23395 q^{-101} +1684 q^{-102} -15056 q^{-103} -19946 q^{-104} -16215 q^{-105} -7040 q^{-106} +2517 q^{-107} +7576 q^{-108} +8655 q^{-109} +5620 q^{-110} +985 q^{-111} -1979 q^{-112} -3381 q^{-113} -2854 q^{-114} -1311 q^{-115} -33 q^{-116} +1072 q^{-117} +1289 q^{-118} +675 q^{-119} +178 q^{-120} -253 q^{-121} -344 q^{-122} -232 q^{-123} -197 q^{-124} +20 q^{-125} +141 q^{-126} +92 q^{-127} +39 q^{-128} -24 q^{-129} -21 q^{-130} +5 q^{-131} -26 q^{-132} -10 q^{-133} +14 q^{-134} +8 q^{-135} +5 q^{-136} -9 q^{-137} -2 q^{-138} +4 q^{-139} - q^{-140}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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