K11a81

K11a80

K11a82

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a81 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,13,19,14 X20,16,21,15 X8,17,9,18 X6,20,7,19 X16,22,17,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5
Dowker-Thistlethwaite code	4 10 12 14 22 2 18 20 8 6 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a81/ThurstonBennequinNumber Hyperbolic Volume 16.051 A-Polynomial See Data:K11a81/A-polynomial

[edit Notes for K11a81's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for K11a81's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 127, 2 }
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-13q^{4}+18q^{3}-20q^{2}+20q-17+13q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+6z^{2}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+5az^{9}+12z^{9}a^{-1}+7z^{9}a^{-3}+4a^{2}z^{8}+15z^{8}a^{-2}+11z^{8}a^{-4}+8z^{8}+a^{3}z^{7}-13az^{7}-26z^{7}a^{-1}-z^{7}a^{-3}+11z^{7}a^{-5}-14a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+8z^{6}a^{-6}-42z^{6}-3a^{3}z^{5}+3az^{5}-23z^{5}a^{-3}-13z^{5}a^{-5}+4z^{5}a^{-7}+15a^{2}z^{4}+39z^{4}a^{-2}+2z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+44z^{4}+3a^{3}z^{3}+9az^{3}+18z^{3}a^{-1}+19z^{3}a^{-3}+5z^{3}a^{-5}-2z^{3}a^{-7}-5a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-4az-6za^{-1}-4za^{-3}-za^{-5}+1$
The A2 invariant	$-q^{12}+q^{10}+q^{8}-q^{6}+3q^{4}-3q^{2}+1+q^{-2}-2q^{-4}+5q^{-6}-3q^{-8}+3q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	$q^{60}-3q^{58}+9q^{56}-19q^{54}+28q^{52}-32q^{50}+15q^{48}+29q^{46}-92q^{44}+157q^{42}-186q^{40}+140q^{38}-15q^{36}-174q^{34}+360q^{32}-449q^{30}+392q^{28}-169q^{26}-152q^{24}+452q^{22}-607q^{20}+545q^{18}-277q^{16}-87q^{14}+395q^{12}-526q^{10}+425q^{8}-144q^{6}-183q^{4}+416q^{2}-444+250q^{-2}+81q^{-4}-420q^{-6}+618q^{-8}-586q^{-10}+326q^{-12}+84q^{-14}-495q^{-16}+758q^{-18}-764q^{-20}+514q^{-22}-89q^{-24}-345q^{-26}+628q^{-28}-662q^{-30}+448q^{-32}-89q^{-34}-249q^{-36}+431q^{-38}-389q^{-40}+158q^{-42}+136q^{-44}-356q^{-46}+405q^{-48}-270q^{-50}+10q^{-52}+254q^{-54}-425q^{-56}+448q^{-58}-321q^{-60}+111q^{-62}+110q^{-64}-276q^{-66}+338q^{-68}-310q^{-70}+215q^{-72}-84q^{-74}-35q^{-76}+125q^{-78}-169q^{-80}+165q^{-82}-127q^{-84}+72q^{-86}-16q^{-88}-29q^{-90}+53q^{-92}-62q^{-94}+51q^{-96}-31q^{-98}+15q^{-100}+q^{-102}-8q^{-104}+10q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

Further Quantum Invariants

K11a81 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["K11a81"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 127, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+8q^{5}-13q^{4}+18q^{3}-20q^{2}+20q-17+13q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+5az^{9}+12z^{9}a^{-1}+7z^{9}a^{-3}+4a^{2}z^{8}+15z^{8}a^{-2}+11z^{8}a^{-4}+8z^{8}+a^{3}z^{7}-13az^{7}-26z^{7}a^{-1}-z^{7}a^{-3}+11z^{7}a^{-5}-14a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+8z^{6}a^{-6}-42z^{6}-3a^{3}z^{5}+3az^{5}-23z^{5}a^{-3}-13z^{5}a^{-5}+4z^{5}a^{-7}+15a^{2}z^{4}+39z^{4}a^{-2}+2z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+44z^{4}+3a^{3}z^{3}+9az^{3}+18z^{3}a^{-1}+19z^{3}a^{-3}+5z^{3}a^{-5}-2z^{3}a^{-7}-5a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-4az-6za^{-1}-4za^{-3}-za^{-5}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a282,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a282,}

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["K11a81"];

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]=

{ $-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$, $q^{7}-4q^{6}+8q^{5}-13q^{4}+18q^{3}-20q^{2}+20q-17+13q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$ }

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]=

{K11a282,}

In[6]:=

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

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

Out[6]=

{K11a282,}

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $0$ $0$ $0$ $0$ $0$ $32$ $32$ $0$ $0$ $0$ $0$ $0$ $32$ $176$ $-160$ $16$ $-48$

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=$2 is the signature of K11a81. 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 5 6 χ 15                       1 1 13                     3   -3 11                   5 1   4 9                 8 3     -5 7               10 5       5 5             10 8         -2 3           10 10           0 1         8 11             3 -1       5 9               -4 -3     3 8                 5 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a80 K11a82