K11a160

K11a159

K11a161

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a160 at Knotilus!

Knot presentations

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

A Braid Representative
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:K11a160/ThurstonBennequinNumber Hyperbolic Volume 17.0985 A-Polynomial See Data:K11a160/A-polynomial

[edit Notes for K11a160'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 K11a160's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+17t^{2}-30t+37-30t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 145, 0 }
Jones polynomial	$q^{6}-4q^{5}+9q^{4}-15q^{3}+20q^{2}-23q+24-20q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+10z^{4}-3a^{2}z^{2}-8z^{2}a^{-2}+2z^{2}a^{-4}+9z^{2}-a^{2}-3a^{-2}+a^{-4}+4$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+7az^{9}+13z^{9}a^{-1}+6z^{9}a^{-3}+10a^{2}z^{8}+15z^{8}a^{-2}+7z^{8}a^{-4}+18z^{8}+8a^{3}z^{7}-az^{7}-17z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-16a^{2}z^{6}-47z^{6}a^{-2}-15z^{6}a^{-4}+z^{6}a^{-6}-51z^{6}+a^{5}z^{5}-12a^{3}z^{5}-17az^{5}-11z^{5}a^{-1}-16z^{5}a^{-3}-9z^{5}a^{-5}-5a^{4}z^{4}+12a^{2}z^{4}+44z^{4}a^{-2}+9z^{4}a^{-4}-2z^{4}a^{-6}+50z^{4}-a^{5}z^{3}+6a^{3}z^{3}+19az^{3}+24z^{3}a^{-1}+18z^{3}a^{-3}+6z^{3}a^{-5}+a^{4}z^{2}-6a^{2}z^{2}-18z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-20z^{2}-2a^{3}z-6az-8za^{-1}-6za^{-3}-2za^{-5}+a^{2}+3a^{-2}+a^{-4}+4$
The A2 invariant	$-q^{14}+2q^{12}-3q^{10}+2q^{8}+q^{6}-3q^{4}+6q^{2}-3+4q^{-2}-q^{-4}-2q^{-6}+3q^{-8}-4q^{-10}+2q^{-12}-q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+16q^{72}-16q^{70}+7q^{68}+16q^{66}-46q^{64}+84q^{62}-113q^{60}+111q^{58}-73q^{56}-15q^{54}+143q^{52}-275q^{50}+376q^{48}-386q^{46}+258q^{44}-5q^{42}-327q^{40}+628q^{38}-780q^{36}+692q^{34}-355q^{32}-146q^{30}+635q^{28}-907q^{26}+852q^{24}-467q^{22}-87q^{20}+565q^{18}-770q^{16}+591q^{14}-104q^{12}-456q^{10}+848q^{8}-849q^{6}+439q^{4}+234q^{2}-898+1280q^{-2}-1218q^{-4}+713q^{-6}+58q^{-8}-812q^{-10}+1298q^{-12}-1328q^{-14}+916q^{-16}-225q^{-18}-485q^{-20}+928q^{-22}-967q^{-24}+608q^{-26}-22q^{-28}-518q^{-30}+780q^{-32}-647q^{-34}+179q^{-36}+404q^{-38}-847q^{-40}+945q^{-42}-667q^{-44}+124q^{-46}+452q^{-48}-845q^{-50}+935q^{-52}-705q^{-54}+288q^{-56}+144q^{-58}-455q^{-60}+556q^{-62}-473q^{-64}+285q^{-66}-69q^{-68}-91q^{-70}+170q^{-72}-173q^{-74}+127q^{-76}-66q^{-78}+18q^{-80}+13q^{-82}-25q^{-84}+22q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

Further Quantum Invariants

K11a160 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["K11a160"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+17t^{2}-30t+37-30t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 145, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+9q^{4}-15q^{3}+20q^{2}-23q+24-20q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+10z^{4}-3a^{2}z^{2}-8z^{2}a^{-2}+2z^{2}a^{-4}+9z^{2}-a^{2}-3a^{-2}+a^{-4}+4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+7az^{9}+13z^{9}a^{-1}+6z^{9}a^{-3}+10a^{2}z^{8}+15z^{8}a^{-2}+7z^{8}a^{-4}+18z^{8}+8a^{3}z^{7}-az^{7}-17z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-16a^{2}z^{6}-47z^{6}a^{-2}-15z^{6}a^{-4}+z^{6}a^{-6}-51z^{6}+a^{5}z^{5}-12a^{3}z^{5}-17az^{5}-11z^{5}a^{-1}-16z^{5}a^{-3}-9z^{5}a^{-5}-5a^{4}z^{4}+12a^{2}z^{4}+44z^{4}a^{-2}+9z^{4}a^{-4}-2z^{4}a^{-6}+50z^{4}-a^{5}z^{3}+6a^{3}z^{3}+19az^{3}+24z^{3}a^{-1}+18z^{3}a^{-3}+6z^{3}a^{-5}+a^{4}z^{2}-6a^{2}z^{2}-18z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-20z^{2}-2a^{3}z-6az-8za^{-1}-6za^{-3}-2za^{-5}+a^{2}+3a^{-2}+a^{-4}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a76, K11a289,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a76, K11a289,}

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["K11a160"];

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]=

{ $t^{4}-6t^{3}+17t^{2}-30t+37-30t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$, $q^{6}-4q^{5}+9q^{4}-15q^{3}+20q^{2}-23q+24-20q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ }

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]=

{K11a76, K11a289,}

In[6]:=

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

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

Out[6]=

{K11a76, K11a289,}

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 $0$ $-8$ $0$ $-16$ $-8$ $0$ ${\frac {112}{3}}$ ${\frac {64}{3}}$ $24$ $0$ $32$ $0$ $0$ $104$ $-104$ ${\frac {632}{3}}$ $-{\frac {88}{3}}$ $40$

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 K11a160. 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                   6 1   5 7                 9 3     -6 5               11 6       5 3             12 9         -3 1           12 11           1 -1         9 13             4 -3       6 11               -5 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a159 K11a161