10 60

10_59

10_61

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 60's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 60 at Knotilus!

Two figure 8 knots on a loop, interlinked.

Knotscape

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20
Gauss code	1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8
Dowker-Thistlethwaite code	4 8 10 14 2 16 18 6 20 12
Conway Notation	[211,211,2]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 12, width is 5, Braid index is 5		[{2, 13}, {1, 10}, {12, 6}, {13, 11}, {9, 3}, {10, 8}, {7, 9}, {8, 12}, {5, 2}, {6, 4}, {3, 5}, {4, 7}, {11, 1}]

[edit Notes on presentations of 10 60]

Computer Talk[show]

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["10 60"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 16 18 6 20 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[211,211,2]

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]=

${\textrm {BR}}(5,\{-1,2,-1,2,2,-3,2,-3,-2,-4,3,-4\})$

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]=

{ 5, 12, 5 }

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[{2, 13}, {1, 10}, {12, 6}, {13, 11}, {9, 3}, {10, 8}, {7, 9}, {8, 12}, {5, 2}, {6, 4}, {3, 5}, {4, 7}, {11, 1}]

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 Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-5] Hyperbolic Volume 13.98 A-Polynomial See Data:10 60/A-polynomial

[edit Notes for 10 60'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 10 60's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-20t+29-20t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 85, 0 }
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-11q+14-14q^{-1}+13q^{-2}-10q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-3z^{2}a^{4}-3a^{4}+3z^{4}a^{2}+6z^{2}a^{2}+4a^{2}-z^{6}-3z^{4}-5z^{2}-2+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+3a^{4}z^{8}+8a^{2}z^{8}+5z^{8}+3a^{5}z^{7}+10a^{3}z^{7}+16az^{7}+9z^{7}a^{-1}+a^{6}z^{6}-3a^{4}z^{6}-7a^{2}z^{6}+8z^{6}a^{-2}+5z^{6}-9a^{5}z^{5}-32a^{3}z^{5}-38az^{5}-11z^{5}a^{-1}+4z^{5}a^{-3}-3a^{6}z^{4}-8a^{4}z^{4}-17a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}-22z^{4}+9a^{5}z^{3}+27a^{3}z^{3}+25az^{3}+5z^{3}a^{-1}-2z^{3}a^{-3}+3a^{6}z^{2}+11a^{4}z^{2}+18a^{2}z^{2}+4z^{2}a^{-2}+14z^{2}-3a^{5}z-7a^{3}z-6az-2za^{-1}-a^{6}-3a^{4}-4a^{2}-a^{-2}-2$
The A2 invariant	$q^{20}+q^{18}-2q^{16}-3q^{10}+3q^{8}+q^{4}+2q^{2}-2+3q^{-2}-3q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}$
The G2 invariant	$q^{94}-2q^{92}+6q^{90}-10q^{88}+12q^{86}-11q^{84}+22q^{80}-45q^{78}+69q^{76}-72q^{74}+47q^{72}+7q^{70}-83q^{68}+155q^{66}-189q^{64}+162q^{62}-75q^{60}-59q^{58}+184q^{56}-259q^{54}+248q^{52}-149q^{50}-2q^{48}+141q^{46}-220q^{44}+196q^{42}-90q^{40}-48q^{38}+160q^{36}-186q^{34}+112q^{32}+35q^{30}-189q^{28}+289q^{26}-278q^{24}+162q^{22}+33q^{20}-231q^{18}+361q^{16}-373q^{14}+267q^{12}-75q^{10}-130q^{8}+270q^{6}-303q^{4}+226q^{2}-75-81q^{-2}+172q^{-4}-168q^{-6}+73q^{-8}+61q^{-10}-170q^{-12}+207q^{-14}-146q^{-16}+18q^{-18}+122q^{-20}-225q^{-22}+252q^{-24}-194q^{-26}+83q^{-28}+41q^{-30}-136q^{-32}+177q^{-34}-161q^{-36}+108q^{-38}-38q^{-40}-20q^{-42}+53q^{-44}-68q^{-46}+57q^{-48}-35q^{-50}+17q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

Further Quantum Invariants[show]

Further quantum knot invariants for 10_60.

A1 Invariants.

Weight	Invariant
1	$q^{13}-2q^{11}+3q^{9}-4q^{7}+3q^{5}-q^{3}+3q^{-1}-3q^{-3}+4q^{-5}-3q^{-7}+q^{-9}$
2	$q^{38}-2q^{36}-2q^{34}+8q^{32}-4q^{30}-12q^{28}+20q^{26}+q^{24}-30q^{22}+24q^{20}+16q^{18}-38q^{16}+12q^{14}+26q^{12}-25q^{10}-7q^{8}+20q^{6}+2q^{4}-20q^{2}+3+29q^{-2}-23q^{-4}-17q^{-6}+39q^{-8}-13q^{-10}-24q^{-12}+28q^{-14}-2q^{-16}-15q^{-18}+9q^{-20}+q^{-22}-3q^{-24}+q^{-26}$
3	$q^{75}-2q^{73}-2q^{71}+3q^{69}+8q^{67}-4q^{65}-19q^{63}+2q^{61}+35q^{59}+10q^{57}-56q^{55}-36q^{53}+74q^{51}+77q^{49}-79q^{47}-130q^{45}+61q^{43}+187q^{41}-22q^{39}-224q^{37}-39q^{35}+239q^{33}+105q^{31}-228q^{29}-159q^{27}+186q^{25}+200q^{23}-132q^{21}-216q^{19}+68q^{17}+212q^{15}-3q^{13}-185q^{11}-66q^{9}+151q^{7}+127q^{5}-96q^{3}-183q+32q^{-1}+223q^{-3}+37q^{-5}-239q^{-7}-105q^{-9}+230q^{-11}+154q^{-13}-187q^{-15}-181q^{-17}+136q^{-19}+177q^{-21}-80q^{-23}-150q^{-25}+38q^{-27}+105q^{-29}-8q^{-31}-68q^{-33}-3q^{-35}+38q^{-37}+2q^{-39}-14q^{-41}-3q^{-43}+6q^{-45}+q^{-47}-3q^{-49}+q^{-51}$
4	$q^{124}-2q^{122}-2q^{120}+3q^{118}+3q^{116}+8q^{114}-11q^{112}-19q^{110}+2q^{108}+17q^{106}+53q^{104}-13q^{102}-80q^{100}-55q^{98}+18q^{96}+191q^{94}+87q^{92}-146q^{90}-262q^{88}-158q^{86}+363q^{84}+436q^{82}+32q^{80}-526q^{78}-711q^{76}+212q^{74}+881q^{72}+712q^{70}-382q^{68}-1405q^{66}-552q^{64}+848q^{62}+1571q^{60}+454q^{58}-1559q^{56}-1531q^{54}+65q^{52}+1887q^{50}+1500q^{48}-920q^{46}-1987q^{44}-950q^{42}+1434q^{40}+2035q^{38}+23q^{36}-1731q^{34}-1557q^{32}+643q^{30}+1904q^{28}+747q^{26}-1098q^{24}-1650q^{22}-125q^{20}+1387q^{18}+1223q^{16}-330q^{14}-1451q^{12}-878q^{10}+624q^{8}+1548q^{6}+608q^{4}-963q^{2}-1604-430q^{-2}+1540q^{-4}+1568q^{-6}-71q^{-8}-1901q^{-10}-1520q^{-12}+929q^{-14}+2007q^{-16}+961q^{-18}-1423q^{-20}-2014q^{-22}+1563q^{-26}+1455q^{-28}-508q^{-30}-1607q^{-32}-557q^{-34}+679q^{-36}+1159q^{-38}+104q^{-40}-801q^{-42}-489q^{-44}+82q^{-46}+562q^{-48}+199q^{-50}-247q^{-52}-206q^{-54}-64q^{-56}+174q^{-58}+86q^{-60}-51q^{-62}-41q^{-64}-35q^{-66}+35q^{-68}+18q^{-70}-10q^{-72}-2q^{-74}-6q^{-76}+6q^{-78}+q^{-80}-3q^{-82}+q^{-84}$
5	$q^{185}-2q^{183}-2q^{181}+3q^{179}+3q^{177}+3q^{175}+q^{173}-11q^{171}-19q^{169}+2q^{167}+26q^{165}+35q^{163}+21q^{161}-37q^{159}-96q^{157}-74q^{155}+49q^{153}+178q^{151}+192q^{149}+14q^{147}-285q^{145}-436q^{143}-199q^{141}+354q^{139}+784q^{137}+623q^{135}-217q^{133}-1192q^{131}-1379q^{129}-298q^{127}+1450q^{125}+2397q^{123}+1412q^{121}-1197q^{119}-3475q^{117}-3188q^{115}+99q^{113}+4143q^{111}+5399q^{109}+2083q^{107}-3824q^{105}-7540q^{103}-5277q^{101}+2111q^{99}+8914q^{97}+8920q^{95}+1106q^{93}-8786q^{91}-12301q^{89}-5470q^{87}+6927q^{85}+14531q^{83}+10158q^{81}-3404q^{79}-15031q^{77}-14349q^{75}-1132q^{73}+13740q^{71}+17219q^{69}+5794q^{67}-10917q^{65}-18399q^{63}-9867q^{61}+7313q^{59}+17977q^{57}+12704q^{55}-3602q^{53}-16274q^{51}-14229q^{49}+318q^{47}+13932q^{45}+14556q^{43}+2264q^{41}-11369q^{39}-14124q^{37}-4186q^{35}+8885q^{33}+13315q^{31}+5752q^{29}-6516q^{27}-12480q^{25}-7229q^{23}+4096q^{21}+11575q^{19}+8978q^{17}-1323q^{15}-10563q^{13}-10941q^{11}-1993q^{9}+9010q^{7}+12920q^{5}+5974q^{3}-6650q-14461q^{-1}-10268q^{-3}+3254q^{-5}+14958q^{-7}+14366q^{-9}+1074q^{-11}-14016q^{-13}-17544q^{-15}-5758q^{-17}+11455q^{-19}+19058q^{-21}+10134q^{-23}-7620q^{-25}-18599q^{-27}-13299q^{-29}+3218q^{-31}+16217q^{-33}+14720q^{-35}+880q^{-37}-12534q^{-39}-14220q^{-41}-3941q^{-43}+8410q^{-45}+12217q^{-47}+5483q^{-49}-4658q^{-51}-9318q^{-53}-5678q^{-55}+1833q^{-57}+6393q^{-59}+4845q^{-61}-169q^{-63}-3844q^{-65}-3594q^{-67}-599q^{-69}+2057q^{-71}+2369q^{-73}+716q^{-75}-970q^{-77}-1356q^{-79}-574q^{-81}+372q^{-83}+713q^{-85}+381q^{-87}-149q^{-89}-329q^{-91}-184q^{-93}+31q^{-95}+137q^{-97}+92q^{-99}-6q^{-101}-61q^{-103}-32q^{-105}+9q^{-107}+15q^{-109}+6q^{-111}+2q^{-113}-5q^{-115}-6q^{-117}+6q^{-119}+q^{-121}-3q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}+q^{18}-2q^{16}-3q^{10}+3q^{8}+q^{4}+2q^{2}-2+3q^{-2}-3q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{52}+q^{50}-q^{48}-5q^{46}-2q^{44}+6q^{42}+5q^{40}-6q^{38}-6q^{36}+11q^{34}+12q^{32}-12q^{30}-15q^{28}+10q^{26}+11q^{24}-13q^{22}-12q^{20}+15q^{18}+10q^{16}-11q^{14}-q^{12}+9q^{10}-6q^{8}-2q^{6}+9q^{4}-6q^{2}-10+11q^{-2}+15q^{-4}-17q^{-6}-12q^{-8}+20q^{-10}+10q^{-12}-18q^{-14}-5q^{-16}+14q^{-18}+3q^{-20}-8q^{-22}-3q^{-24}+5q^{-26}-2q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}-2q^{38}+2q^{36}+3q^{34}-9q^{32}+7q^{30}+6q^{28}-20q^{26}+14q^{24}+11q^{22}-28q^{20}+17q^{18}+14q^{16}-27q^{14}+8q^{12}+13q^{10}-12q^{8}-5q^{6}+7q^{4}+11q^{2}-11-6q^{-2}+27q^{-4}-15q^{-6}-17q^{-8}+30q^{-10}-11q^{-12}-17q^{-14}+22q^{-16}-4q^{-18}-10q^{-20}+9q^{-22}-3q^{-26}+q^{-28}$
1,0,0	$q^{27}+q^{25}+q^{23}-2q^{21}-3q^{17}-3q^{13}+4q^{11}+3q^{7}+q^{5}+q^{3}-2q^{-1}+2q^{-3}-3q^{-5}+2q^{-7}-q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$

B2 Invariants.

Weight	Invariant
0,1	$q^{40}-2q^{38}+6q^{36}-9q^{34}+15q^{32}-21q^{30}+26q^{28}-32q^{26}+32q^{24}-31q^{22}+22q^{20}-13q^{18}-2q^{16}+19q^{14}-34q^{12}+51q^{10}-58q^{8}+65q^{6}-61q^{4}+55q^{2}-43+26q^{-2}-9q^{-4}-7q^{-6}+19q^{-8}-28q^{-10}+33q^{-12}-33q^{-14}+30q^{-16}-24q^{-18}+18q^{-20}-11q^{-22}+6q^{-24}-3q^{-26}+q^{-28}$
1,0	$q^{66}-2q^{62}-2q^{60}+4q^{58}+6q^{56}-3q^{54}-12q^{52}-4q^{50}+16q^{48}+15q^{46}-12q^{44}-27q^{42}-q^{40}+32q^{38}+19q^{36}-25q^{34}-31q^{32}+9q^{30}+36q^{28}+7q^{26}-30q^{24}-17q^{22}+19q^{20}+20q^{18}-13q^{16}-20q^{14}+7q^{12}+21q^{10}-4q^{8}-22q^{6}-q^{4}+25q^{2}+10-23q^{-2}-17q^{-4}+22q^{-6}+27q^{-8}-12q^{-10}-34q^{-12}-3q^{-14}+33q^{-16}+19q^{-18}-22q^{-20}-29q^{-22}+5q^{-24}+26q^{-26}+9q^{-28}-14q^{-30}-14q^{-32}+3q^{-34}+10q^{-36}+3q^{-38}-3q^{-40}-3q^{-42}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{94}-2q^{92}+6q^{90}-10q^{88}+12q^{86}-11q^{84}+22q^{80}-45q^{78}+69q^{76}-72q^{74}+47q^{72}+7q^{70}-83q^{68}+155q^{66}-189q^{64}+162q^{62}-75q^{60}-59q^{58}+184q^{56}-259q^{54}+248q^{52}-149q^{50}-2q^{48}+141q^{46}-220q^{44}+196q^{42}-90q^{40}-48q^{38}+160q^{36}-186q^{34}+112q^{32}+35q^{30}-189q^{28}+289q^{26}-278q^{24}+162q^{22}+33q^{20}-231q^{18}+361q^{16}-373q^{14}+267q^{12}-75q^{10}-130q^{8}+270q^{6}-303q^{4}+226q^{2}-75-81q^{-2}+172q^{-4}-168q^{-6}+73q^{-8}+61q^{-10}-170q^{-12}+207q^{-14}-146q^{-16}+18q^{-18}+122q^{-20}-225q^{-22}+252q^{-24}-194q^{-26}+83q^{-28}+41q^{-30}-136q^{-32}+177q^{-34}-161q^{-36}+108q^{-38}-38q^{-40}-20q^{-42}+53q^{-44}-68q^{-46}+57q^{-48}-35q^{-50}+17q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

.

Computer Talk[show]

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["10 60"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+7t^{2}-20t+29-20t^{-1}+7t^{-2}-t^{-3}$

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]=

{ 85, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{4}-4q^{3}+8q^{2}-11q+14-14q^{-1}+13q^{-2}-10q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{6}-3z^{2}a^{4}-3a^{4}+3z^{4}a^{2}+6z^{2}a^{2}+4a^{2}-z^{6}-3z^{4}-5z^{2}-2+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{9}+az^{9}+3a^{4}z^{8}+8a^{2}z^{8}+5z^{8}+3a^{5}z^{7}+10a^{3}z^{7}+16az^{7}+9z^{7}a^{-1}+a^{6}z^{6}-3a^{4}z^{6}-7a^{2}z^{6}+8z^{6}a^{-2}+5z^{6}-9a^{5}z^{5}-32a^{3}z^{5}-38az^{5}-11z^{5}a^{-1}+4z^{5}a^{-3}-3a^{6}z^{4}-8a^{4}z^{4}-17a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}-22z^{4}+9a^{5}z^{3}+27a^{3}z^{3}+25az^{3}+5z^{3}a^{-1}-2z^{3}a^{-3}+3a^{6}z^{2}+11a^{4}z^{2}+18a^{2}z^{2}+4z^{2}a^{-2}+14z^{2}-3a^{5}z-7a^{3}z-6az-2za^{-1}-a^{6}-3a^{4}-4a^{2}-a^{-2}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n165,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_86,}

Computer Talk[show]

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["10 60"];

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]=

{ $-t^{3}+7t^{2}-20t+29-20t^{-1}+7t^{-2}-t^{-3}$, $q^{4}-4q^{3}+8q^{2}-11q+14-14q^{-1}+13q^{-2}-10q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ }

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]=

{K11n165,}

In[6]:=

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

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

Out[6]=

{10_86,}

Vassiliev invariants

V2 and V3:

(-1, 1)

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 $-4$ $8$ $8$ $-{\frac {62}{3}}$ $-{\frac {10}{3}}$ $-32$ ${\frac {80}{3}}$ ${\frac {32}{3}}$ $8$ $-{\frac {32}{3}}$ $32$ ${\frac {248}{3}}$ ${\frac {40}{3}}$ ${\frac {2609}{30}}$ ${\frac {1382}{15}}$ $-{\frac {3062}{45}}$ $-{\frac {17}{18}}$ $-{\frac {271}{30}}$

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 $t^{r}q^{j}$ 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or $j-2r=s-1$, where $s=$0 is the signature of 10 60. 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 4 χ 9                     1 1 7                   3   -3 5                 5 1   4 3               6 3     -3 1             8 5       3 -1           7 7         0 -3         6 7           -1 -5       4 7             3 -7     2 6               -4 -9   1 4                 3 -11   2                   -2 -13 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{12}-4q^{11}+4q^{10}+9q^{9}-28q^{8}+17q^{7}+39q^{6}-80q^{5}+28q^{4}+91q^{3}-136q^{2}+22q+143-162q^{-1}-q^{-2}+165q^{-3}-144q^{-4}-28q^{-5}+147q^{-6}-93q^{-7}-42q^{-8}+97q^{-9}-39q^{-10}-34q^{-11}+43q^{-12}-8q^{-13}-15q^{-14}+11q^{-15}-3q^{-17}+q^{-18}$
3	$q^{24}-4q^{23}+4q^{22}+5q^{21}-8q^{20}-15q^{19}+20q^{18}+41q^{17}-49q^{16}-80q^{15}+80q^{14}+154q^{13}-116q^{12}-268q^{11}+150q^{10}+411q^{9}-157q^{8}-585q^{7}+144q^{6}+752q^{5}-81q^{4}-920q^{3}+10q^{2}+1028q+105-1111q^{-1}-205q^{-2}+1115q^{-3}+328q^{-4}-1087q^{-5}-422q^{-6}+996q^{-7}+510q^{-8}-872q^{-9}-566q^{-10}+712q^{-11}+594q^{-12}-540q^{-13}-580q^{-14}+367q^{-15}+525q^{-16}-207q^{-17}-446q^{-18}+89q^{-19}+340q^{-20}-5q^{-21}-237q^{-22}-37q^{-23}+149q^{-24}+46q^{-25}-81q^{-26}-40q^{-27}+39q^{-28}+26q^{-29}-15q^{-30}-15q^{-31}+6q^{-32}+5q^{-33}-3q^{-35}+q^{-36}$
4	$q^{40}-4q^{39}+4q^{38}+5q^{37}-12q^{36}+5q^{35}-12q^{34}+32q^{33}+22q^{32}-82q^{31}-q^{30}-22q^{29}+169q^{28}+110q^{27}-320q^{26}-143q^{25}-63q^{24}+615q^{23}+473q^{22}-800q^{21}-714q^{20}-375q^{19}+1520q^{18}+1528q^{17}-1280q^{16}-1950q^{15}-1425q^{14}+2619q^{13}+3491q^{12}-1172q^{11}-3513q^{10}-3439q^{9}+3210q^{8}+5875q^{7}-126q^{6}-4591q^{5}-5888q^{4}+2829q^{3}+7705q^{2}+1513q-4619-7858q^{-1}+1655q^{-2}+8346q^{-3}+3084q^{-4}-3679q^{-5}-8782q^{-6}+153q^{-7}+7773q^{-8}+4205q^{-9}-2126q^{-10}-8618q^{-11}-1359q^{-12}+6248q^{-13}+4757q^{-14}-281q^{-15}-7461q^{-16}-2620q^{-17}+4048q^{-18}+4583q^{-19}+1473q^{-20}-5449q^{-21}-3221q^{-22}+1664q^{-23}+3546q^{-24}+2540q^{-25}-3029q^{-26}-2834q^{-27}-158q^{-28}+1950q^{-29}+2512q^{-30}-1016q^{-31}-1717q^{-32}-881q^{-33}+550q^{-34}+1659q^{-35}+7q^{-36}-623q^{-37}-712q^{-38}-119q^{-39}+736q^{-40}+192q^{-41}-65q^{-42}-308q^{-43}-192q^{-44}+215q^{-45}+88q^{-46}+51q^{-47}-75q^{-48}-88q^{-49}+42q^{-50}+15q^{-51}+26q^{-52}-8q^{-53}-22q^{-54}+6q^{-55}+5q^{-57}-3q^{-59}+q^{-60}$
5	$q^{60}-4q^{59}+4q^{58}+5q^{57}-12q^{56}+q^{55}+8q^{54}+13q^{52}-q^{51}-53q^{50}-28q^{49}+63q^{48}+98q^{47}+58q^{46}-107q^{45}-268q^{44}-173q^{43}+243q^{42}+628q^{41}+390q^{40}-448q^{39}-1214q^{38}-955q^{37}+629q^{36}+2314q^{35}+2043q^{34}-760q^{33}-3870q^{32}-3950q^{31}+379q^{30}+5989q^{29}+7057q^{28}+788q^{27}-8430q^{26}-11461q^{25}-3261q^{24}+10649q^{23}+17198q^{22}+7522q^{21}-12237q^{20}-23812q^{19}-13540q^{18}+12335q^{17}+30612q^{16}+21362q^{15}-10740q^{14}-36811q^{13}-30057q^{12}+7035q^{11}+41591q^{10}+39116q^{9}-1816q^{8}-44414q^{7}-47270q^{6}-4751q^{5}+45119q^{4}+54206q^{3}+11476q^{2}-43822q-58974-18273q^{-1}+40926q^{-2}+62017q^{-3}+24100q^{-4}-36876q^{-5}-62884q^{-6}-29276q^{-7}+31978q^{-8}+62395q^{-9}+33340q^{-10}-26575q^{-11}-60287q^{-12}-36755q^{-13}+20653q^{-14}+57144q^{-15}+39304q^{-16}-14307q^{-17}-52724q^{-18}-41185q^{-19}+7582q^{-20}+47206q^{-21}+42059q^{-22}-674q^{-23}-40432q^{-24}-41809q^{-25}-6032q^{-26}+32659q^{-27}+40014q^{-28}+11998q^{-29}-24126q^{-30}-36536q^{-31}-16696q^{-32}+15479q^{-33}+31482q^{-34}+19480q^{-35}-7415q^{-36}-25111q^{-37}-20175q^{-38}+607q^{-39}+18265q^{-40}+18798q^{-41}+4212q^{-42}-11549q^{-43}-15802q^{-44}-6997q^{-45}+5868q^{-46}+11967q^{-47}+7727q^{-48}-1657q^{-49}-7988q^{-50}-7003q^{-51}-935q^{-52}+4579q^{-53}+5464q^{-54}+2059q^{-55}-2081q^{-56}-3687q^{-57}-2191q^{-58}+535q^{-59}+2177q^{-60}+1772q^{-61}+197q^{-62}-1078q^{-63}-1206q^{-64}-412q^{-65}+429q^{-66}+691q^{-67}+384q^{-68}-103q^{-69}-366q^{-70}-251q^{-71}-q^{-72}+138q^{-73}+147q^{-74}+48q^{-75}-67q^{-76}-73q^{-77}-15q^{-78}+9q^{-79}+24q^{-80}+26q^{-81}-8q^{-82}-15q^{-83}-q^{-84}+5q^{-87}-3q^{-89}+q^{-90}$

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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