K11a106

K11a105

K11a107

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a106 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a106/ThurstonBennequinNumber Hyperbolic Volume 13.4001 A-Polynomial See Data:K11a106/A-polynomial

[edit Notes for K11a106's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for K11a106's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 93, 0 }
Jones polynomial	$q^{6}-3q^{5}+5q^{4}-9q^{3}+13q^{2}-14q+15-13q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}+14z^{4}-5a^{2}z^{2}-12z^{2}a^{-2}+3z^{2}a^{-4}+15z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+3az^{9}+6z^{9}a^{-1}+3z^{9}a^{-3}+4a^{2}z^{8}+6z^{8}a^{-2}+4z^{8}a^{-4}+6z^{8}+4a^{3}z^{7}-2az^{7}-13z^{7}a^{-1}-4z^{7}a^{-3}+3z^{7}a^{-5}+3a^{4}z^{6}-4a^{2}z^{6}-27z^{6}a^{-2}-12z^{6}a^{-4}+z^{6}a^{-6}-21z^{6}+a^{5}z^{5}-5a^{3}z^{5}+7z^{5}a^{-1}-9z^{5}a^{-3}-10z^{5}a^{-5}-6a^{4}z^{4}+a^{2}z^{4}+39z^{4}a^{-2}+10z^{4}a^{-4}-3z^{4}a^{-6}+33z^{4}-2a^{5}z^{3}-a^{3}z^{3}+az^{3}+9z^{3}a^{-1}+17z^{3}a^{-3}+8z^{3}a^{-5}+3a^{4}z^{2}-4a^{2}z^{2}-22z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}-23z^{2}+a^{5}z+a^{3}z-2az-6za^{-1}-6za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$
The A2 invariant	$-q^{14}+q^{12}-2q^{10}+q^{8}+q^{6}-q^{4}+4q^{2}-2+3q^{-2}+2q^{-8}-3q^{-10}-q^{-14}+q^{-18}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-6q^{70}-2q^{68}+16q^{66}-29q^{64}+41q^{62}-42q^{60}+27q^{58}-2q^{56}-37q^{54}+74q^{52}-101q^{50}+102q^{48}-79q^{46}+25q^{44}+47q^{42}-117q^{40}+170q^{38}-174q^{36}+125q^{34}-38q^{32}-72q^{30}+154q^{28}-185q^{26}+155q^{24}-59q^{22}-46q^{20}+126q^{18}-135q^{16}+69q^{14}+37q^{12}-137q^{10}+180q^{8}-143q^{6}+34q^{4}+117q^{2}-236+292q^{-2}-242q^{-4}+106q^{-6}+61q^{-8}-211q^{-10}+291q^{-12}-270q^{-14}+173q^{-16}-22q^{-18}-114q^{-20}+200q^{-22}-191q^{-24}+99q^{-26}+17q^{-28}-118q^{-30}+151q^{-32}-107q^{-34}+q^{-36}+114q^{-38}-186q^{-40}+195q^{-42}-128q^{-44}-5q^{-46}+121q^{-48}-198q^{-50}+207q^{-52}-152q^{-54}+59q^{-56}+37q^{-58}-105q^{-60}+135q^{-62}-120q^{-64}+77q^{-66}-23q^{-68}-18q^{-70}+42q^{-72}-48q^{-74}+41q^{-76}-25q^{-78}+11q^{-80}+2q^{-82}-8q^{-84}+7q^{-86}-7q^{-88}+4q^{-90}-2q^{-92}+q^{-94}$

Further Quantum Invariants

K11a106 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["K11a106"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-3q^{5}+5q^{4}-9q^{3}+13q^{2}-14q+15-13q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}+14z^{4}-5a^{2}z^{2}-12z^{2}a^{-2}+3z^{2}a^{-4}+15z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+3az^{9}+6z^{9}a^{-1}+3z^{9}a^{-3}+4a^{2}z^{8}+6z^{8}a^{-2}+4z^{8}a^{-4}+6z^{8}+4a^{3}z^{7}-2az^{7}-13z^{7}a^{-1}-4z^{7}a^{-3}+3z^{7}a^{-5}+3a^{4}z^{6}-4a^{2}z^{6}-27z^{6}a^{-2}-12z^{6}a^{-4}+z^{6}a^{-6}-21z^{6}+a^{5}z^{5}-5a^{3}z^{5}+7z^{5}a^{-1}-9z^{5}a^{-3}-10z^{5}a^{-5}-6a^{4}z^{4}+a^{2}z^{4}+39z^{4}a^{-2}+10z^{4}a^{-4}-3z^{4}a^{-6}+33z^{4}-2a^{5}z^{3}-a^{3}z^{3}+az^{3}+9z^{3}a^{-1}+17z^{3}a^{-3}+8z^{3}a^{-5}+3a^{4}z^{2}-4a^{2}z^{2}-22z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}-23z^{2}+a^{5}z+a^{3}z-2az-6za^{-1}-6za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a194, K11a346,}

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["K11a106"];

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]=

{ $t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$, $q^{6}-3q^{5}+5q^{4}-9q^{3}+13q^{2}-14q+15-13q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ }

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]=

{K11a194, K11a346,}

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, -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 $4$ $-8$ $8$ $-{\frac {34}{3}}$ $-{\frac {38}{3}}$ $-32$ ${\frac {16}{3}}$ ${\frac {64}{3}}$ $24$ ${\frac {32}{3}}$ $32$ $-{\frac {136}{3}}$ $-{\frac {152}{3}}$ $-{\frac {209}{30}}$ $-{\frac {662}{15}}$ ${\frac {1862}{45}}$ $-{\frac {463}{18}}$ ${\frac {751}{30}}$

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 $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of K11a106. 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 13                       1 1 11                     2   -2 9                   3 1   2 7                 6 2     -4 5               7 3       4 3             7 6         -1 1           8 7           1 -1         6 8             2 -3       4 7               -3 -5     2 6                 4 -7   1 4                   -3 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a105 K11a107