K11a62

K11a61

K11a63

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a62 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,15,1,16 X6,17,7,18 X12,19,13,20 X14,21,15,22
Gauss code	1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 16 10 2 18 20 22 6 12 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a62/ThurstonBennequinNumber Hyperbolic Volume 10.6386 A-Polynomial See Data:K11a62/A-polynomial

[edit Notes for K11a62's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 6

[edit Notes for K11a62's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-8t^{2}+9t-9+9t^{-1}-8t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}+2z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, -6 }
Jones polynomial	$q^{-1}-2q^{-2}+4q^{-3}-5q^{-4}+7q^{-5}-8q^{-6}+8q^{-7}-7q^{-8}+6q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{10}-4z^{2}a^{10}-3a^{10}+2z^{6}a^{8}+10z^{4}a^{8}+14z^{2}a^{8}+6a^{8}-z^{8}a^{6}-6z^{6}a^{6}-12z^{4}a^{6}-11z^{2}a^{6}-5a^{6}+z^{6}a^{4}+5z^{4}a^{4}+7z^{2}a^{4}+3a^{4}$
Kauffman polynomial (db, data sources)	$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-z^{2}a^{14}+3z^{5}a^{13}-2z^{3}a^{13}+za^{13}+4z^{6}a^{12}-6z^{4}a^{12}+4z^{2}a^{12}+4z^{7}a^{11}-7z^{5}a^{11}+2z^{3}a^{11}+4z^{8}a^{10}-11z^{6}a^{10}+10z^{4}a^{10}-9z^{2}a^{10}+3a^{10}+3z^{9}a^{9}-9z^{7}a^{9}+6z^{5}a^{9}-3z^{3}a^{9}+z^{10}a^{8}+2z^{8}a^{8}-24z^{6}a^{8}+41z^{4}a^{8}-27z^{2}a^{8}+6a^{8}+5z^{9}a^{7}-24z^{7}a^{7}+34z^{5}a^{7}-17z^{3}a^{7}+3za^{7}+z^{10}a^{6}-z^{8}a^{6}-15z^{6}a^{6}+35z^{4}a^{6}-23z^{2}a^{6}+5a^{6}+2z^{9}a^{5}-11z^{7}a^{5}+18z^{5}a^{5}-9z^{3}a^{5}+za^{5}+z^{8}a^{4}-6z^{6}a^{4}+12z^{4}a^{4}-10z^{2}a^{4}+3a^{4}$
The A2 invariant	$-q^{36}-q^{34}-q^{30}+q^{28}+2q^{24}+q^{22}-q^{20}+q^{18}-2q^{16}+q^{14}+q^{10}+q^{8}+q^{4}$
The G2 invariant	$q^{196}-q^{194}+2q^{192}-2q^{190}+q^{188}-2q^{184}+4q^{182}-5q^{180}+6q^{178}-6q^{176}+3q^{174}+2q^{172}-6q^{170}+11q^{168}-12q^{166}+10q^{164}-8q^{162}-q^{160}+7q^{158}-14q^{156}+15q^{154}-12q^{152}+5q^{150}-6q^{146}+8q^{144}-8q^{142}+6q^{140}-6q^{138}+4q^{136}-5q^{134}+3q^{132}+2q^{130}-8q^{128}+14q^{126}-13q^{124}+7q^{122}+q^{120}-10q^{118}+11q^{116}-4q^{114}-5q^{112}+14q^{110}-14q^{108}+7q^{106}+11q^{104}-24q^{102}+31q^{100}-26q^{98}+12q^{96}+7q^{94}-21q^{92}+33q^{90}-30q^{88}+23q^{86}-9q^{84}-7q^{82}+19q^{80}-24q^{78}+18q^{76}-9q^{74}-6q^{72}+16q^{70}-18q^{68}+7q^{66}+10q^{64}-25q^{62}+29q^{60}-22q^{58}+q^{56}+20q^{54}-34q^{52}+38q^{50}-26q^{48}+9q^{46}+11q^{44}-22q^{42}+25q^{40}-17q^{38}+10q^{36}-4q^{32}+6q^{30}-5q^{28}+4q^{26}-q^{24}+q^{22}$

Further Quantum Invariants

K11a62 Quantum Invariants.

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["K11a62"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-8t^{2}+9t-9+9t^{-1}-8t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}+2z^{4}+6z^{2}+1$

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]=

{ 55, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-1}-2q^{-2}+4q^{-3}-5q^{-4}+7q^{-5}-8q^{-6}+8q^{-7}-7q^{-8}+6q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{10}-4z^{2}a^{10}-3a^{10}+2z^{6}a^{8}+10z^{4}a^{8}+14z^{2}a^{8}+6a^{8}-z^{8}a^{6}-6z^{6}a^{6}-12z^{4}a^{6}-11z^{2}a^{6}-5a^{6}+z^{6}a^{4}+5z^{4}a^{4}+7z^{2}a^{4}+3a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-z^{2}a^{14}+3z^{5}a^{13}-2z^{3}a^{13}+za^{13}+4z^{6}a^{12}-6z^{4}a^{12}+4z^{2}a^{12}+4z^{7}a^{11}-7z^{5}a^{11}+2z^{3}a^{11}+4z^{8}a^{10}-11z^{6}a^{10}+10z^{4}a^{10}-9z^{2}a^{10}+3a^{10}+3z^{9}a^{9}-9z^{7}a^{9}+6z^{5}a^{9}-3z^{3}a^{9}+z^{10}a^{8}+2z^{8}a^{8}-24z^{6}a^{8}+41z^{4}a^{8}-27z^{2}a^{8}+6a^{8}+5z^{9}a^{7}-24z^{7}a^{7}+34z^{5}a^{7}-17z^{3}a^{7}+3za^{7}+z^{10}a^{6}-z^{8}a^{6}-15z^{6}a^{6}+35z^{4}a^{6}-23z^{2}a^{6}+5a^{6}+2z^{9}a^{5}-11z^{7}a^{5}+18z^{5}a^{5}-9z^{3}a^{5}+za^{5}+z^{8}a^{4}-6z^{6}a^{4}+12z^{4}a^{4}-10z^{2}a^{4}+3a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a62"];

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]=

{ $-t^{4}+5t^{3}-8t^{2}+9t-9+9t^{-1}-8t^{-2}+5t^{-3}-t^{-4}$, $q^{-1}-2q^{-2}+4q^{-3}-5q^{-4}+7q^{-5}-8q^{-6}+8q^{-7}-7q^{-8}+6q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$ }

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]=

{}

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:

(6, -17)

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 $24$ $-136$ $288$ $876$ $124$ $-3264$ $-{\frac {18832}{3}}$ $-{\frac {3232}{3}}$ $-776$ $2304$ $9248$ $21024$ $2976$ ${\frac {231671}{5}}$ ${\frac {28148}{15}}$ ${\frac {84188}{5}}$ $339$ ${\frac {10391}{5}}$

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 $j-2r=s+1$ or $j-2r=s-1$, where $s=$-6 is the signature of K11a62. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -1                       1 1 -3                     1   -1 -5                   3 1   2 -7                 3 2     -1 -9               4 2       2 -11             4 3         -1 -13           4 4           0 -15         3 4             1 -17       3 4               -1 -19     1 3                 2 -21   1 3                   -2 -23   1                     1 -25 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-7$ $i=-5$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a61 K11a63