K11a167

K11a166

K11a168

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a167 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a167/ThurstonBennequinNumber Hyperbolic Volume 15.5032 A-Polynomial See Data:K11a167/A-polynomial

[edit Notes for K11a167's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11a167's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+11t^{2}-26t+35-26t^{-1}+11t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 113, 0 }
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-12q^{3}+16q^{2}-18q+18-15q^{-1}+11q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}-a^{4}z^{2}+4a^{2}z^{2}-z^{2}a^{-2}+z^{2}a^{-4}-3z^{2}-a^{4}+3a^{2}+a^{-2}-2$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+4az^{9}+10z^{9}a^{-1}+6z^{9}a^{-3}+4a^{2}z^{8}+6z^{8}a^{-2}+7z^{8}a^{-4}+3z^{8}+4a^{3}z^{7}-4az^{7}-26z^{7}a^{-1}-14z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-27z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-9z^{6}+a^{5}z^{5}-4a^{3}z^{5}+5az^{5}+28z^{5}a^{-1}+8z^{5}a^{-3}-10z^{5}a^{-5}-6a^{4}z^{4}-10a^{2}z^{4}+27z^{4}a^{-2}+15z^{4}a^{-4}-2z^{4}a^{-6}+6z^{4}-2a^{5}z^{3}-2a^{3}z^{3}-6az^{3}-12z^{3}a^{-1}-2z^{3}a^{-3}+4z^{3}a^{-5}+4a^{4}z^{2}+9a^{2}z^{2}-6z^{2}a^{-2}-4z^{2}a^{-4}+3z^{2}+a^{5}z+2a^{3}z+2az+2za^{-1}+za^{-3}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	Data:K11a167/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a167/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a167 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["K11a167"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+11t^{2}-26t+35-26t^{-1}+11t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 113, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+8q^{4}-12q^{3}+16q^{2}-18q+18-15q^{-1}+11q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+4az^{9}+10z^{9}a^{-1}+6z^{9}a^{-3}+4a^{2}z^{8}+6z^{8}a^{-2}+7z^{8}a^{-4}+3z^{8}+4a^{3}z^{7}-4az^{7}-26z^{7}a^{-1}-14z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-27z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-9z^{6}+a^{5}z^{5}-4a^{3}z^{5}+5az^{5}+28z^{5}a^{-1}+8z^{5}a^{-3}-10z^{5}a^{-5}-6a^{4}z^{4}-10a^{2}z^{4}+27z^{4}a^{-2}+15z^{4}a^{-4}-2z^{4}a^{-6}+6z^{4}-2a^{5}z^{3}-2a^{3}z^{3}-6az^{3}-12z^{3}a^{-1}-2z^{3}a^{-3}+4z^{3}a^{-5}+4a^{4}z^{2}+9a^{2}z^{2}-6z^{2}a^{-2}-4z^{2}a^{-4}+3z^{2}+a^{5}z+2a^{3}z+2az+2za^{-1}+za^{-3}-a^{4}-3a^{2}-a^{-2}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a11,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a11,}

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["K11a167"];

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]=

{ $-2t^{3}+11t^{2}-26t+35-26t^{-1}+11t^{-2}-2t^{-3}$, $q^{6}-4q^{5}+8q^{4}-12q^{3}+16q^{2}-18q+18-15q^{-1}+11q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ }

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]=

{K11a11,}

In[6]:=

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

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

Out[6]=

{K11a11,}

Vassiliev invariants

V2 and V3:

(0, -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 $0$ $-8$ $0$ $16$ $8$ $0$ ${\frac {16}{3}}$ ${\frac {160}{3}}$ $-40$ $0$ $32$ $0$ $0$ $56$ $88$ $-{\frac {344}{3}}$ ${\frac {136}{3}}$ $-24$

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 K11a167. 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                     3   -3 9                   5 1   4 7                 7 3     -4 5               9 5       4 3             9 7         -2 1           9 9           0 -1         7 10             3 -3       4 8               -4 -5     2 7                 5 -7   1 4                   -3 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a166 K11a168