K11a118

K11a117

K11a119

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a118 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a118/ThurstonBennequinNumber Hyperbolic Volume 13.3497 A-Polynomial See Data:K11a118/A-polynomial

[edit Notes for K11a118's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a118's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-10t^{2}+20t-23+20t^{-1}-10t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, 2 }
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+10q^{5}-12q^{4}+14q^{3}-14q^{2}+11q-8+5q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}+4z^{2}a^{-4}-2z^{2}a^{-6}-5z^{2}+2a^{2}-a^{-2}+3a^{-4}-a^{-6}-2$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+2z^{8}a^{-2}+6z^{8}a^{-4}+6z^{8}a^{-6}+2z^{8}+2az^{7}-3z^{7}a^{-1}-19z^{7}a^{-3}-9z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-9z^{6}a^{-2}-26z^{6}a^{-4}-17z^{6}a^{-6}+3z^{6}a^{-8}-2z^{6}-6az^{5}+3z^{5}a^{-1}+31z^{5}a^{-3}+10z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+13z^{4}a^{-2}+43z^{4}a^{-4}+21z^{4}a^{-6}-6z^{4}a^{-8}-7z^{4}+4az^{3}-9z^{3}a^{-1}-23z^{3}a^{-3}-2z^{3}a^{-5}+6z^{3}a^{-7}-2z^{3}a^{-9}+5a^{2}z^{2}-10z^{2}a^{-2}-24z^{2}a^{-4}-10z^{2}a^{-6}+z^{2}a^{-8}+8z^{2}+5za^{-1}+7za^{-3}+za^{-5}-za^{-7}-2a^{2}+a^{-2}+3a^{-4}+a^{-6}-2$
The A2 invariant	Data:K11a118/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a118/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a118 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["K11a118"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-10t^{2}+20t-23+20t^{-1}-10t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 87, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-6q^{6}+10q^{5}-12q^{4}+14q^{3}-14q^{2}+11q-8+5q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+2z^{8}a^{-2}+6z^{8}a^{-4}+6z^{8}a^{-6}+2z^{8}+2az^{7}-3z^{7}a^{-1}-19z^{7}a^{-3}-9z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-9z^{6}a^{-2}-26z^{6}a^{-4}-17z^{6}a^{-6}+3z^{6}a^{-8}-2z^{6}-6az^{5}+3z^{5}a^{-1}+31z^{5}a^{-3}+10z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+13z^{4}a^{-2}+43z^{4}a^{-4}+21z^{4}a^{-6}-6z^{4}a^{-8}-7z^{4}+4az^{3}-9z^{3}a^{-1}-23z^{3}a^{-3}-2z^{3}a^{-5}+6z^{3}a^{-7}-2z^{3}a^{-9}+5a^{2}z^{2}-10z^{2}a^{-2}-24z^{2}a^{-4}-10z^{2}a^{-6}+z^{2}a^{-8}+8z^{2}+5za^{-1}+7za^{-3}+za^{-5}-za^{-7}-2a^{2}+a^{-2}+3a^{-4}+a^{-6}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a90,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a45,}

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["K11a118"];

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]=

{ $2t^{3}-10t^{2}+20t-23+20t^{-1}-10t^{-2}+2t^{-3}$, $-q^{8}+3q^{7}-6q^{6}+10q^{5}-12q^{4}+14q^{3}-14q^{2}+11q-8+5q^{-1}-2q^{-2}+q^{-3}$ }

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]=

{K11a90,}

In[6]:=

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

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

Out[6]=

{K11a45,}

Vassiliev invariants

V2 and V3:

(-2, 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 $-8$ $8$ $32$ ${\frac {164}{3}}$ ${\frac {4}{3}}$ $-64$ $-{\frac {400}{3}}$ $-{\frac {352}{3}}$ $8$ $-{\frac {256}{3}}$ $32$ $-{\frac {1312}{3}}$ $-{\frac {32}{3}}$ $-{\frac {8431}{15}}$ $-{\frac {7276}{15}}$ ${\frac {9596}{45}}$ $-{\frac {257}{9}}$ ${\frac {1169}{15}}$

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 K11a118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a117 K11a119