9 28

9_27

9_29

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 28's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 28 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X15,18,16,1 X9,16,10,17 X17,10,18,11 X7283
Gauss code	-1, 9, -2, 1, -4, 5, -9, 2, -7, 8, -3, 4, -5, 3, -6, 7, -8, 6
Dowker-Thistlethwaite code	4 8 12 2 16 14 6 18 10
Conway Notation	[21,21,2+]

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

[edit Notes on presentations of 9 28]

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 28"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X15,18,16,1 X9,16,10,17 X17,10,18,11 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 12 2 16 14 6 18 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[21,21,2+]

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]=

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

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

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 [-9][-2] Hyperbolic Volume 11.5632 A-Polynomial See Data:9 28/A-polynomial

[edit Notes for 9 28's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for 9 28's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-2} + t^{-3}$
Conway polynomial	$z^6+z^4+z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, -2 }
Jones polynomial	$-q^2+3 q-5+8 q^{-1} -8 q^{-2} +9 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^2 a^6+a^6-2 z^4 a^4-5 z^2 a^4-4 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+5 a^2-z^4-2 z^2-1$
Kauffman polynomial (db, data sources)	$z^4 a^8-z^2 a^8+3 z^5 a^7-4 z^3 a^7+2 z a^7+4 z^6 a^6-4 z^4 a^6+2 z^2 a^6-a^6+3 z^7 a^5+2 z^5 a^5-9 z^3 a^5+6 z a^5+z^8 a^4+8 z^6 a^4-17 z^4 a^4+12 z^2 a^4-4 a^4+6 z^7 a^3-5 z^5 a^3-7 z^3 a^3+6 z a^3+z^8 a^2+7 z^6 a^2-19 z^4 a^2+14 z^2 a^2-5 a^2+3 z^7 a-3 z^5 a-4 z^3 a+3 z a+3 z^6-7 z^4+5 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1}$
The A2 invariant	$q^{22}-q^{18}+q^{16}-3 q^{14}-q^{12}+4 q^6+3 q^2- q^{-2} + q^{-4} - q^{-6}$
The G2 invariant	$q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-3 q^{104}-4 q^{102}+13 q^{100}-20 q^{98}+27 q^{96}-25 q^{94}+15 q^{92}+6 q^{90}-29 q^{88}+54 q^{86}-62 q^{84}+53 q^{82}-29 q^{80}-10 q^{78}+49 q^{76}-72 q^{74}+73 q^{72}-44 q^{70}+3 q^{68}+32 q^{66}-53 q^{64}+37 q^{62}-7 q^{60}-33 q^{58}+56 q^{56}-58 q^{54}+23 q^{52}+30 q^{50}-81 q^{48}+105 q^{46}-99 q^{44}+53 q^{42}+8 q^{40}-67 q^{38}+103 q^{36}-103 q^{34}+79 q^{32}-24 q^{30}-28 q^{28}+63 q^{26}-64 q^{24}+43 q^{22}-34 q^{18}+52 q^{16}-35 q^{14}+5 q^{12}+41 q^{10}-70 q^8+77 q^6-52 q^4+5 q^2+39-70 q^{-2} +76 q^{-4} -56 q^{-6} +24 q^{-8} +8 q^{-10} -33 q^{-12} +39 q^{-14} -32 q^{-16} +19 q^{-18} -5 q^{-20} -5 q^{-22} +7 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 9_28.

A1 Invariants.

Weight	Invariant
1	$q^{15}-2 q^{13}+2 q^{11}-3 q^9+q^7+q^5+3 q-2 q^{-1} +2 q^{-3} - q^{-5}$
2	$q^{42}-2 q^{40}-q^{38}+6 q^{36}-6 q^{34}-4 q^{32}+15 q^{30}-8 q^{28}-11 q^{26}+18 q^{24}-3 q^{22}-13 q^{20}+8 q^{18}+4 q^{16}-6 q^{14}-5 q^{12}+9 q^{10}+3 q^8-14 q^6+9 q^4+12 q^2-16+3 q^{-2} +13 q^{-4} -10 q^{-6} -3 q^{-8} +7 q^{-10} -2 q^{-12} -2 q^{-14} + q^{-16}$
3	$q^{81}-2 q^{79}-q^{77}+3 q^{75}+3 q^{73}-6 q^{71}-7 q^{69}+14 q^{67}+12 q^{65}-20 q^{63}-25 q^{61}+27 q^{59}+41 q^{57}-31 q^{55}-61 q^{53}+28 q^{51}+77 q^{49}-13 q^{47}-88 q^{45}-2 q^{43}+86 q^{41}+18 q^{39}-68 q^{37}-36 q^{35}+49 q^{33}+41 q^{31}-21 q^{29}-49 q^{27}-2 q^{25}+46 q^{23}+28 q^{21}-47 q^{19}-48 q^{17}+42 q^{15}+65 q^{13}-31 q^{11}-77 q^9+18 q^7+84 q^5+3 q^3-83 q-17 q^{-1} +68 q^{-3} +35 q^{-5} -51 q^{-7} -39 q^{-9} +31 q^{-11} +37 q^{-13} -12 q^{-15} -28 q^{-17} - q^{-19} +17 q^{-21} +3 q^{-23} -7 q^{-25} -3 q^{-27} +2 q^{-29} +2 q^{-31} - q^{-33}$
4	$q^{132}-2 q^{130}-q^{128}+3 q^{126}+3 q^{122}-9 q^{120}-2 q^{118}+15 q^{116}+2 q^{114}+2 q^{112}-37 q^{110}-12 q^{108}+52 q^{106}+34 q^{104}+6 q^{102}-110 q^{100}-67 q^{98}+104 q^{96}+136 q^{94}+65 q^{92}-218 q^{90}-218 q^{88}+96 q^{86}+285 q^{84}+234 q^{82}-252 q^{80}-413 q^{78}-53 q^{76}+339 q^{74}+441 q^{72}-118 q^{70}-474 q^{68}-249 q^{66}+208 q^{64}+488 q^{62}+91 q^{60}-318 q^{58}-331 q^{56}-12 q^{54}+338 q^{52}+226 q^{50}-75 q^{48}-277 q^{46}-179 q^{44}+117 q^{42}+270 q^{40}+139 q^{38}-185 q^{36}-288 q^{34}-77 q^{32}+285 q^{30}+306 q^{28}-87 q^{26}-357 q^{24}-255 q^{22}+259 q^{20}+433 q^{18}+52 q^{16}-347 q^{14}-415 q^{12}+125 q^{10}+454 q^8+228 q^6-200 q^4-477 q^2-81+315 q^{-2} +323 q^{-4} +27 q^{-6} -353 q^{-8} -210 q^{-10} +81 q^{-12} +246 q^{-14} +165 q^{-16} -137 q^{-18} -169 q^{-20} -65 q^{-22} +83 q^{-24} +134 q^{-26} + q^{-28} -53 q^{-30} -63 q^{-32} -7 q^{-34} +46 q^{-36} +16 q^{-38} + q^{-40} -17 q^{-42} -10 q^{-44} +7 q^{-46} +3 q^{-48} +3 q^{-50} -2 q^{-52} -2 q^{-54} + q^{-56}$
5	$q^{195}-2 q^{193}-q^{191}+3 q^{189}-4 q^{181}-q^{179}+11 q^{177}+5 q^{175}-11 q^{173}-19 q^{171}-12 q^{169}+18 q^{167}+47 q^{165}+39 q^{163}-34 q^{161}-107 q^{159}-80 q^{157}+47 q^{155}+184 q^{153}+187 q^{151}-26 q^{149}-320 q^{147}-368 q^{145}-32 q^{143}+450 q^{141}+644 q^{139}+219 q^{137}-573 q^{135}-1018 q^{133}-544 q^{131}+601 q^{129}+1424 q^{127}+1036 q^{125}-455 q^{123}-1776 q^{121}-1638 q^{119}+85 q^{117}+1977 q^{115}+2238 q^{113}+458 q^{111}-1899 q^{109}-2695 q^{107}-1115 q^{105}+1552 q^{103}+2912 q^{101}+1698 q^{99}-994 q^{97}-2790 q^{95}-2114 q^{93}+328 q^{91}+2378 q^{89}+2303 q^{87}+276 q^{85}-1794 q^{83}-2190 q^{81}-786 q^{79}+1118 q^{77}+1950 q^{75}+1119 q^{73}-514 q^{71}-1563 q^{69}-1322 q^{67}-45 q^{65}+1228 q^{63}+1443 q^{61}+470 q^{59}-897 q^{57}-1548 q^{55}-862 q^{53}+655 q^{51}+1664 q^{49}+1205 q^{47}-426 q^{45}-1801 q^{43}-1574 q^{41}+191 q^{39}+1909 q^{37}+1948 q^{35}+121 q^{33}-1950 q^{31}-2308 q^{29}-525 q^{27}+1848 q^{25}+2601 q^{23}+1002 q^{21}-1535 q^{19}-2749 q^{17}-1518 q^{15}+1055 q^{13}+2680 q^{11}+1945 q^9-409 q^7-2337 q^5-2241 q^3-257 q+1801 q^{-1} +2235 q^{-3} +841 q^{-5} -1092 q^{-7} -1992 q^{-9} -1223 q^{-11} +417 q^{-13} +1515 q^{-15} +1320 q^{-17} +150 q^{-19} -954 q^{-21} -1167 q^{-23} -488 q^{-25} +436 q^{-27} +862 q^{-29} +581 q^{-31} -62 q^{-33} -507 q^{-35} -498 q^{-37} -146 q^{-39} +229 q^{-41} +337 q^{-43} +181 q^{-45} -51 q^{-47} -172 q^{-49} -146 q^{-51} -28 q^{-53} +74 q^{-55} +83 q^{-57} +33 q^{-59} -18 q^{-61} -34 q^{-63} -25 q^{-65} - q^{-67} +17 q^{-69} +10 q^{-71} -3 q^{-75} -3 q^{-77} -3 q^{-79} +2 q^{-81} +2 q^{-83} - q^{-85}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{18}+q^{16}-3 q^{14}-q^{12}+4 q^6+3 q^2- q^{-2} + q^{-4} - q^{-6}$
1,1	$q^{60}-4 q^{58}+10 q^{56}-20 q^{54}+38 q^{52}-66 q^{50}+100 q^{48}-146 q^{46}+198 q^{44}-246 q^{42}+284 q^{40}-300 q^{38}+289 q^{36}-230 q^{34}+132 q^{32}-4 q^{30}-149 q^{28}+306 q^{26}-448 q^{24}+552 q^{22}-611 q^{20}+608 q^{18}-560 q^{16}+454 q^{14}-320 q^{12}+166 q^{10}-2 q^8-126 q^6+239 q^4-296 q^2+326-308 q^{-2} +262 q^{-4} -208 q^{-6} +148 q^{-8} -96 q^{-10} +54 q^{-12} -28 q^{-14} +12 q^{-16} -4 q^{-18} + q^{-20}$
2,0	$q^{56}-2 q^{52}-q^{50}+2 q^{48}+q^{46}-4 q^{44}+7 q^{40}-q^{38}-7 q^{36}+2 q^{34}+8 q^{32}-6 q^{28}+5 q^{26}+q^{24}-9 q^{22}-3 q^{20}+q^{18}-5 q^{16}-3 q^{14}+8 q^{12}+3 q^{10}-2 q^8+5 q^6+11 q^4-2 q^2-6+5 q^{-2} +3 q^{-4} -5 q^{-6} -3 q^{-8} +2 q^{-10} +2 q^{-12} -2 q^{-14} - q^{-16} + q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-2 q^{46}+4 q^{42}-7 q^{40}-q^{38}+11 q^{36}-8 q^{34}-q^{32}+17 q^{30}-8 q^{28}-5 q^{26}+7 q^{24}-9 q^{22}-10 q^{20}-3 q^{18}+3 q^{16}-2 q^{12}+14 q^{10}+9 q^8-10 q^6+10 q^4+6 q^2-13+5 q^{-2} +3 q^{-4} -8 q^{-6} +3 q^{-8} + q^{-10} -2 q^{-12} + q^{-14}$
1,0,0	$q^{29}+q^{25}-q^{23}+q^{21}-4 q^{19}-q^{17}-3 q^{15}+q^{11}+3 q^9+4 q^7+q^5+3 q^3-q+ q^{-1} -2 q^{-3} + q^{-5} - q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}-q^{58}+q^{54}-3 q^{52}-7 q^{50}+7 q^{46}-q^{42}+16 q^{40}+12 q^{38}-4 q^{36}-3 q^{34}+3 q^{32}-13 q^{30}-22 q^{28}-8 q^{26}-4 q^{24}-13 q^{22}+19 q^{18}+4 q^{16}+4 q^{14}+20 q^{12}+13 q^{10}-6 q^8+q^6+8 q^4-3 q^2-10+3 q^{-4} -5 q^{-6} -2 q^{-8} +3 q^{-10} - q^{-14} + q^{-16}$
1,0,0,0	$q^{36}+q^{32}+q^{30}-q^{28}+q^{26}-4 q^{24}-2 q^{22}-3 q^{20}-3 q^{18}+q^{14}+4 q^{12}+3 q^{10}+5 q^8+q^6+3 q^4-q^2-2 q^{-4} + q^{-6} - q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-2 q^{46}+4 q^{44}-6 q^{42}+9 q^{40}-11 q^{38}+13 q^{36}-14 q^{34}+11 q^{32}-9 q^{30}+2 q^{28}+3 q^{26}-11 q^{24}+17 q^{22}-22 q^{20}+25 q^{18}-25 q^{16}+24 q^{14}-18 q^{12}+14 q^{10}-5 q^8+8 q^4-10 q^2+13-13 q^{-2} +13 q^{-4} -10 q^{-6} +7 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-14}$
1,0	$q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+5 q^{68}-8 q^{64}-6 q^{62}+6 q^{60}+12 q^{58}+q^{56}-13 q^{54}-6 q^{52}+11 q^{50}+14 q^{48}-4 q^{46}-13 q^{44}-q^{42}+11 q^{40}+2 q^{38}-12 q^{36}-8 q^{34}+5 q^{32}+5 q^{30}-7 q^{28}-9 q^{26}+3 q^{24}+10 q^{22}-7 q^{18}+q^{16}+14 q^{14}+7 q^{12}-9 q^{10}-10 q^8+9 q^6+15 q^4-14-7 q^{-2} +9 q^{-4} +10 q^{-6} -4 q^{-8} -9 q^{-10} -2 q^{-12} +5 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-2 q^{64}+2 q^{62}-3 q^{60}+5 q^{58}-8 q^{56}+6 q^{54}-8 q^{52}+12 q^{50}-10 q^{48}+10 q^{46}-6 q^{44}+12 q^{42}-q^{40}+q^{36}-9 q^{34}+8 q^{32}-21 q^{30}+8 q^{28}-25 q^{26}+17 q^{24}-19 q^{22}+19 q^{20}-13 q^{18}+20 q^{16}-q^{14}+13 q^{12}+q^{10}+q^8+8 q^6-7 q^4+7 q^2-12+10 q^{-2} -11 q^{-4} +8 q^{-6} -9 q^{-8} +6 q^{-10} -4 q^{-12} +3 q^{-14} -2 q^{-16} + q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-3 q^{104}-4 q^{102}+13 q^{100}-20 q^{98}+27 q^{96}-25 q^{94}+15 q^{92}+6 q^{90}-29 q^{88}+54 q^{86}-62 q^{84}+53 q^{82}-29 q^{80}-10 q^{78}+49 q^{76}-72 q^{74}+73 q^{72}-44 q^{70}+3 q^{68}+32 q^{66}-53 q^{64}+37 q^{62}-7 q^{60}-33 q^{58}+56 q^{56}-58 q^{54}+23 q^{52}+30 q^{50}-81 q^{48}+105 q^{46}-99 q^{44}+53 q^{42}+8 q^{40}-67 q^{38}+103 q^{36}-103 q^{34}+79 q^{32}-24 q^{30}-28 q^{28}+63 q^{26}-64 q^{24}+43 q^{22}-34 q^{18}+52 q^{16}-35 q^{14}+5 q^{12}+41 q^{10}-70 q^8+77 q^6-52 q^4+5 q^2+39-70 q^{-2} +76 q^{-4} -56 q^{-6} +24 q^{-8} +8 q^{-10} -33 q^{-12} +39 q^{-14} -32 q^{-16} +19 q^{-18} -5 q^{-20} -5 q^{-22} +7 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32}$

. </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 28"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-2} + t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^6+z^4+z^2+1$

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]=

{ 51, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^2+3 q-5+8 q^{-1} -8 q^{-2} +9 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^2 a^6+a^6-2 z^4 a^4-5 z^2 a^4-4 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+5 a^2-z^4-2 z^2-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^4 a^8-z^2 a^8+3 z^5 a^7-4 z^3 a^7+2 z a^7+4 z^6 a^6-4 z^4 a^6+2 z^2 a^6-a^6+3 z^7 a^5+2 z^5 a^5-9 z^3 a^5+6 z a^5+z^8 a^4+8 z^6 a^4-17 z^4 a^4+12 z^2 a^4-4 a^4+6 z^7 a^3-5 z^5 a^3-7 z^3 a^3+6 z a^3+z^8 a^2+7 z^6 a^2-19 z^4 a^2+14 z^2 a^2-5 a^2+3 z^7 a-3 z^5 a-4 z^3 a+3 z a+3 z^6-7 z^4+5 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_29, 10_163, K11n87,}

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 28"];

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]=

{ $t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-2} + t^{-3}$, $-q^2+3 q-5+8 q^{-1} -8 q^{-2} +9 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7}$ }

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]=

{9_29, 10_163, K11n87,}

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 $4$ $0$ $8$ $-\frac{34}{3}$ $-\frac{14}{3}$ $0$ $32$ $0$ $0$ $\frac{32}{3}$ $0$ $-\frac{136}{3}$ $-\frac{56}{3}$ $-\frac{2609}{30}$ $-\frac{622}{15}$ $\frac{1742}{45}$ $-\frac{79}{18}$ $\frac{271}{30}$

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 $s=$-2 is the signature of 9 28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 2   2 1               3 1   -2 -1             5 2     3 -3           4 4       0 -5         5 4         1 -7       3 4           1 -9     2 5             -3 -11   1 3               2 -13   2                 -2 -15 1                   1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

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$n$	$J_n$
2	$q^7-3 q^6+10 q^4-13 q^3-7 q^2+33 q-23-26 q^{-1} +61 q^{-2} -26 q^{-3} -49 q^{-4} +78 q^{-5} -20 q^{-6} -63 q^{-7} +77 q^{-8} -10 q^{-9} -59 q^{-10} +56 q^{-11} -38 q^{-13} +27 q^{-14} +3 q^{-15} -15 q^{-16} +8 q^{-17} + q^{-18} -3 q^{-19} + q^{-20}$
3	$-q^{15}+3 q^{14}-5 q^{12}-5 q^{11}+13 q^{10}+14 q^9-23 q^8-32 q^7+29 q^6+63 q^5-29 q^4-102 q^3+17 q^2+149 q+4-187 q^{-1} -49 q^{-2} +235 q^{-3} +85 q^{-4} -253 q^{-5} -144 q^{-6} +281 q^{-7} +181 q^{-8} -276 q^{-9} -234 q^{-10} +282 q^{-11} +256 q^{-12} -258 q^{-13} -282 q^{-14} +235 q^{-15} +284 q^{-16} -196 q^{-17} -274 q^{-18} +150 q^{-19} +252 q^{-20} -110 q^{-21} -206 q^{-22} +62 q^{-23} +166 q^{-24} -35 q^{-25} -116 q^{-26} +13 q^{-27} +77 q^{-28} -5 q^{-29} -44 q^{-30} - q^{-31} +25 q^{-32} -12 q^{-34} + q^{-35} +4 q^{-36} + q^{-37} -3 q^{-38} + q^{-39}$
4	$q^{26}-3 q^{25}+5 q^{23}+5 q^{21}-20 q^{20}-7 q^{19}+23 q^{18}+15 q^{17}+35 q^{16}-73 q^{15}-63 q^{14}+33 q^{13}+69 q^{12}+168 q^{11}-124 q^{10}-211 q^9-71 q^8+101 q^7+470 q^6-43 q^5-376 q^4-362 q^3-42 q^2+850 q+253-384 q^{-1} -758 q^{-2} -438 q^{-3} +1127 q^{-4} +681 q^{-5} -158 q^{-6} -1087 q^{-7} -978 q^{-8} +1195 q^{-9} +1080 q^{-10} +223 q^{-11} -1261 q^{-12} -1492 q^{-13} +1093 q^{-14} +1350 q^{-15} +616 q^{-16} -1282 q^{-17} -1854 q^{-18} +882 q^{-19} +1453 q^{-20} +940 q^{-21} -1151 q^{-22} -2007 q^{-23} +586 q^{-24} +1355 q^{-25} +1142 q^{-26} -850 q^{-27} -1895 q^{-28} +236 q^{-29} +1036 q^{-30} +1155 q^{-31} -441 q^{-32} -1498 q^{-33} -44 q^{-34} +579 q^{-35} +930 q^{-36} -85 q^{-37} -939 q^{-38} -146 q^{-39} +187 q^{-40} +570 q^{-41} +76 q^{-42} -453 q^{-43} -95 q^{-44} -2 q^{-45} +256 q^{-46} +76 q^{-47} -170 q^{-48} -24 q^{-49} -34 q^{-50} +85 q^{-51} +33 q^{-52} -54 q^{-53} +4 q^{-54} -16 q^{-55} +21 q^{-56} +8 q^{-57} -15 q^{-58} +4 q^{-59} -3 q^{-60} +4 q^{-61} + q^{-62} -3 q^{-63} + q^{-64}$
5	$-q^{40}+3 q^{39}-5 q^{37}+2 q^{34}+13 q^{33}+7 q^{32}-23 q^{31}-24 q^{30}-9 q^{29}+18 q^{28}+64 q^{27}+57 q^{26}-32 q^{25}-126 q^{24}-127 q^{23}-8 q^{22}+185 q^{21}+289 q^{20}+124 q^{19}-234 q^{18}-502 q^{17}-360 q^{16}+176 q^{15}+734 q^{14}+767 q^{13}+47 q^{12}-928 q^{11}-1284 q^{10}-503 q^9+947 q^8+1871 q^7+1217 q^6-733 q^5-2382 q^4-2143 q^3+178 q^2+2771 q+3150+661 q^{-1} -2816 q^{-2} -4201 q^{-3} -1806 q^{-4} +2675 q^{-5} +5078 q^{-6} +3015 q^{-7} -2081 q^{-8} -5826 q^{-9} -4379 q^{-10} +1444 q^{-11} +6292 q^{-12} +5552 q^{-13} -482 q^{-14} -6579 q^{-15} -6752 q^{-16} -339 q^{-17} +6650 q^{-18} +7623 q^{-19} +1345 q^{-20} -6618 q^{-21} -8470 q^{-22} -2104 q^{-23} +6423 q^{-24} +8998 q^{-25} +2976 q^{-26} -6159 q^{-27} -9479 q^{-28} -3621 q^{-29} +5737 q^{-30} +9649 q^{-31} +4343 q^{-32} -5186 q^{-33} -9694 q^{-34} -4894 q^{-35} +4460 q^{-36} +9408 q^{-37} +5392 q^{-38} -3553 q^{-39} -8863 q^{-40} -5726 q^{-41} +2556 q^{-42} +8004 q^{-43} +5788 q^{-44} -1483 q^{-45} -6836 q^{-46} -5651 q^{-47} +506 q^{-48} +5562 q^{-49} +5112 q^{-50} +313 q^{-51} -4144 q^{-52} -4437 q^{-53} -854 q^{-54} +2895 q^{-55} +3532 q^{-56} +1109 q^{-57} -1787 q^{-58} -2657 q^{-59} -1115 q^{-60} +1003 q^{-61} +1809 q^{-62} +971 q^{-63} -466 q^{-64} -1166 q^{-65} -727 q^{-66} +180 q^{-67} +664 q^{-68} +497 q^{-69} -21 q^{-70} -374 q^{-71} -302 q^{-72} -14 q^{-73} +182 q^{-74} +161 q^{-75} +27 q^{-76} -80 q^{-77} -89 q^{-78} -17 q^{-79} +45 q^{-80} +34 q^{-81} -7 q^{-83} -16 q^{-84} -9 q^{-85} +16 q^{-86} +4 q^{-87} -7 q^{-88} + q^{-89} -3 q^{-91} +4 q^{-92} + q^{-93} -3 q^{-94} + q^{-95}$
6	$q^{57}-3 q^{56}+5 q^{54}-7 q^{51}+5 q^{50}-13 q^{49}-7 q^{48}+32 q^{47}+15 q^{46}+9 q^{45}-35 q^{44}-9 q^{43}-69 q^{42}-47 q^{41}+102 q^{40}+117 q^{39}+120 q^{38}-45 q^{37}-50 q^{36}-328 q^{35}-330 q^{34}+84 q^{33}+354 q^{32}+613 q^{31}+332 q^{30}+199 q^{29}-814 q^{28}-1298 q^{27}-729 q^{26}+163 q^{25}+1417 q^{24}+1713 q^{23}+1867 q^{22}-541 q^{21}-2659 q^{20}-3156 q^{19}-2042 q^{18}+945 q^{17}+3462 q^{16}+5874 q^{15}+2618 q^{14}-2128 q^{13}-6111 q^{12}-7151 q^{11}-3450 q^{10}+2593 q^9+10526 q^8+9438 q^7+3327 q^6-5935 q^5-12847 q^4-12319 q^3-4092 q^2+11604 q+17056+13880 q^{-1} +573 q^{-2} -14619 q^{-3} -22321 q^{-4} -16287 q^{-5} +6070 q^{-6} +20900 q^{-7} +25831 q^{-8} +12673 q^{-9} -9865 q^{-10} -28956 q^{-11} -30012 q^{-12} -4947 q^{-13} +18792 q^{-14} +34868 q^{-15} +26341 q^{-16} -113 q^{-17} -30369 q^{-18} -41162 q^{-19} -17520 q^{-20} +12345 q^{-21} +39359 q^{-22} +37834 q^{-23} +10925 q^{-24} -27986 q^{-25} -48207 q^{-26} -28343 q^{-27} +4690 q^{-28} +40377 q^{-29} +45815 q^{-30} +20514 q^{-31} -24123 q^{-32} -51840 q^{-33} -36388 q^{-34} -2294 q^{-35} +39425 q^{-36} +50779 q^{-37} +28108 q^{-38} -19698 q^{-39} -52910 q^{-40} -42170 q^{-41} -8743 q^{-42} +36600 q^{-43} +53162 q^{-44} +34408 q^{-45} -13892 q^{-46} -50924 q^{-47} -45889 q^{-48} -15587 q^{-49} +30618 q^{-50} +52020 q^{-51} +39362 q^{-52} -5668 q^{-53} -44279 q^{-54} -46153 q^{-55} -22440 q^{-56} +20573 q^{-57} +45495 q^{-58} +41019 q^{-59} +4005 q^{-60} -32331 q^{-61} -40782 q^{-62} -26670 q^{-63} +8214 q^{-64} +33253 q^{-65} +36870 q^{-66} +11602 q^{-67} -17701 q^{-68} -29660 q^{-69} -25326 q^{-70} -2067 q^{-71} +18586 q^{-72} +27045 q^{-73} +13711 q^{-74} -5476 q^{-75} -16493 q^{-76} -18597 q^{-77} -6635 q^{-78} +6751 q^{-79} +15478 q^{-80} +10541 q^{-81} +841 q^{-82} -6267 q^{-83} -10291 q^{-84} -5932 q^{-85} +652 q^{-86} +6698 q^{-87} +5643 q^{-88} +2030 q^{-89} -1141 q^{-90} -4205 q^{-91} -3291 q^{-92} -904 q^{-93} +2196 q^{-94} +2111 q^{-95} +1183 q^{-96} +295 q^{-97} -1264 q^{-98} -1271 q^{-99} -667 q^{-100} +603 q^{-101} +537 q^{-102} +379 q^{-103} +321 q^{-104} -288 q^{-105} -361 q^{-106} -268 q^{-107} +176 q^{-108} +84 q^{-109} +55 q^{-110} +138 q^{-111} -53 q^{-112} -78 q^{-113} -78 q^{-114} +65 q^{-115} +2 q^{-116} -11 q^{-117} +40 q^{-118} -11 q^{-119} -10 q^{-120} -19 q^{-121} +23 q^{-122} - q^{-123} -11 q^{-124} +9 q^{-125} -3 q^{-126} -3 q^{-128} +4 q^{-129} + q^{-130} -3 q^{-131} + q^{-132}$
7	$-q^{77}+3 q^{76}-5 q^{74}+7 q^{71}-5 q^{69}+13 q^{68}-2 q^{67}-23 q^{66}-15 q^{65}-9 q^{64}+35 q^{63}+37 q^{62}+3 q^{61}+48 q^{60}-12 q^{59}-93 q^{58}-112 q^{57}-123 q^{56}+62 q^{55}+181 q^{54}+183 q^{53}+291 q^{52}+108 q^{51}-218 q^{50}-472 q^{49}-749 q^{48}-378 q^{47}+200 q^{46}+674 q^{45}+1362 q^{44}+1204 q^{43}+412 q^{42}-758 q^{41}-2361 q^{40}-2630 q^{39}-1670 q^{38}+104 q^{37}+3047 q^{36}+4635 q^{35}+4352 q^{34}+2038 q^{33}-2974 q^{32}-6867 q^{31}-8261 q^{30}-6202 q^{29}+747 q^{28}+8024 q^{27}+13005 q^{26}+13048 q^{25}+4750 q^{24}-6746 q^{23}-17245 q^{22}-21869 q^{21}-14196 q^{20}+967 q^{19}+18712 q^{18}+31318 q^{17}+27719 q^{16}+10541 q^{15}-15230 q^{14}-38842 q^{13}-43597 q^{12}-28150 q^{11}+4392 q^{10}+41509 q^9+59441 q^8+50828 q^7+14529 q^6-36503 q^5-71811 q^4-76194 q^3-41156 q^2+22305 q+77588+100568 q^{-1} +73445 q^{-2} +1668 q^{-3} -74387 q^{-4} -121116 q^{-5} -108325 q^{-6} -33227 q^{-7} +61444 q^{-8} +134199 q^{-9} +142082 q^{-10} +70799 q^{-11} -39263 q^{-12} -139390 q^{-13} -172112 q^{-14} -109851 q^{-15} +10404 q^{-16} +135374 q^{-17} +195649 q^{-18} +148502 q^{-19} +23194 q^{-20} -124698 q^{-21} -212770 q^{-22} -183037 q^{-23} -57485 q^{-24} +108176 q^{-25} +222542 q^{-26} +213190 q^{-27} +91190 q^{-28} -89270 q^{-29} -227473 q^{-30} -237242 q^{-31} -121219 q^{-32} +68997 q^{-33} +227690 q^{-34} +256635 q^{-35} +148013 q^{-36} -50093 q^{-37} -225958 q^{-38} -271062 q^{-39} -170188 q^{-40} +32240 q^{-41} +222248 q^{-42} +282653 q^{-43} +189379 q^{-44} -16678 q^{-45} -218279 q^{-46} -291142 q^{-47} -205389 q^{-48} +1790 q^{-49} +213003 q^{-50} +298140 q^{-51} +220128 q^{-52} +12336 q^{-53} -206785 q^{-54} -302683 q^{-55} -233293 q^{-56} -27729 q^{-57} +197579 q^{-58} +304994 q^{-59} +246035 q^{-60} +44594 q^{-61} -184961 q^{-62} -303315 q^{-63} -256959 q^{-64} -63871 q^{-65} +166889 q^{-66} +296421 q^{-67} +265617 q^{-68} +84992 q^{-69} -143322 q^{-70} -282689 q^{-71} -269630 q^{-72} -106496 q^{-73} +113998 q^{-74} +260755 q^{-75} +267199 q^{-76} +126626 q^{-77} -80406 q^{-78} -231033 q^{-79} -256636 q^{-80} -142071 q^{-81} +45353 q^{-82} +193919 q^{-83} +236775 q^{-84} +151020 q^{-85} -11527 q^{-86} -152801 q^{-87} -208950 q^{-88} -150896 q^{-89} -16897 q^{-90} +110527 q^{-91} +174334 q^{-92} +142131 q^{-93} +38068 q^{-94} -71539 q^{-95} -137113 q^{-96} -125449 q^{-97} -49806 q^{-98} +38658 q^{-99} +100353 q^{-100} +103690 q^{-101} +53101 q^{-102} -14169 q^{-103} -67912 q^{-104} -79898 q^{-105} -49284 q^{-106} -1678 q^{-107} +41674 q^{-108} +57380 q^{-109} +41122 q^{-110} +9779 q^{-111} -22693 q^{-112} -38092 q^{-113} -31149 q^{-114} -12426 q^{-115} +10356 q^{-116} +23492 q^{-117} +21687 q^{-118} +11359 q^{-119} -3443 q^{-120} -13181 q^{-121} -13772 q^{-122} -8967 q^{-123} +39 q^{-124} +6883 q^{-125} +8148 q^{-126} +6185 q^{-127} +1000 q^{-128} -3226 q^{-129} -4318 q^{-130} -3912 q^{-131} -1160 q^{-132} +1349 q^{-133} +2176 q^{-134} +2319 q^{-135} +838 q^{-136} -574 q^{-137} -963 q^{-138} -1200 q^{-139} -520 q^{-140} +149 q^{-141} +357 q^{-142} +692 q^{-143} +311 q^{-144} -112 q^{-145} -153 q^{-146} -284 q^{-147} -102 q^{-148} +3 q^{-149} -13 q^{-150} +175 q^{-151} +93 q^{-152} -39 q^{-153} -22 q^{-154} -58 q^{-155} +8 q^{-156} +7 q^{-157} -40 q^{-158} +37 q^{-159} +24 q^{-160} -12 q^{-161} -4 q^{-162} -13 q^{-163} +13 q^{-164} +6 q^{-165} -16 q^{-166} +5 q^{-167} +5 q^{-168} -3 q^{-169} -3 q^{-171} +4 q^{-172} + q^{-173} -3 q^{-174} + q^{-175}$

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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