K11n160

K11n159

K11n161

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n160 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,11,4,10 X12,6,13,5 X20,8,21,7 X16,10,17,9 X11,19,12,18 X22,13,1,14 X8,16,9,15 X17,4,18,5 X2,19,3,20 X14,21,15,22
Gauss code	1, -10, -2, 9, 3, -1, 4, -8, 5, 2, -6, -3, 7, -11, 8, -5, -9, 6, 10, -4, 11, -7
Dowker-Thistlethwaite code	6 -10 12 20 16 -18 22 8 -4 2 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n160/ThurstonBennequinNumber Hyperbolic Volume 14.3018 A-Polynomial See Data:K11n160/A-polynomial

[edit Notes for K11n160'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 K11n160's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-6t^{2}+16t-21+16t^{-1}-6t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 67, 2 }
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+9q^{5}-11q^{4}+12q^{3}-10q^{2}+8q-5+2q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-4}-3z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}-7z^{2}a^{-2}+8z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+8z^{8}a^{-4}+5z^{8}a^{-6}+z^{7}a^{-1}-3z^{7}a^{-3}+z^{7}a^{-5}+5z^{7}a^{-7}-8z^{6}a^{-2}-24z^{6}a^{-4}-13z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}+6z^{5}a^{-3}-10z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}+17z^{4}a^{-2}+35z^{4}a^{-4}+15z^{4}a^{-6}-6z^{4}a^{-8}+3z^{4}-7z^{3}a^{-1}-4z^{3}a^{-3}+14z^{3}a^{-5}+9z^{3}a^{-7}-2z^{3}a^{-9}-17z^{2}a^{-2}-21z^{2}a^{-4}-8z^{2}a^{-6}+z^{2}a^{-8}-5z^{2}+za^{-1}-4za^{-5}-3za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$
The A2 invariant	$2q^{4}-1+2q^{-2}-3q^{-4}+q^{-6}+3q^{-12}-q^{-14}+3q^{-16}-q^{-18}-2q^{-20}+q^{-22}-q^{-24}$
The G2 invariant	Data:K11n160/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n160 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["K11n160"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-6t^{2}+16t-21+16t^{-1}-6t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 67, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-6q^{6}+9q^{5}-11q^{4}+12q^{3}-10q^{2}+8q-5+2q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+8z^{8}a^{-4}+5z^{8}a^{-6}+z^{7}a^{-1}-3z^{7}a^{-3}+z^{7}a^{-5}+5z^{7}a^{-7}-8z^{6}a^{-2}-24z^{6}a^{-4}-13z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}+6z^{5}a^{-3}-10z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}+17z^{4}a^{-2}+35z^{4}a^{-4}+15z^{4}a^{-6}-6z^{4}a^{-8}+3z^{4}-7z^{3}a^{-1}-4z^{3}a^{-3}+14z^{3}a^{-5}+9z^{3}a^{-7}-2z^{3}a^{-9}-17z^{2}a^{-2}-21z^{2}a^{-4}-8z^{2}a^{-6}+z^{2}a^{-8}-5z^{2}+za^{-1}-4za^{-5}-3za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n7, K11n131,}

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["K11n160"];

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]=

{ $t^{3}-6t^{2}+16t-21+16t^{-1}-6t^{-2}+t^{-3}$, $-q^{8}+3q^{7}-6q^{6}+9q^{5}-11q^{4}+12q^{3}-10q^{2}+8q-5+2q^{-1}$ }

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]=

{K11n7, K11n131,}

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, 3)

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 $4$ $24$ $8$ ${\frac {206}{3}}$ ${\frac {10}{3}}$ $96$ $240$ $-32$ $88$ ${\frac {32}{3}}$ $288$ ${\frac {824}{3}}$ ${\frac {40}{3}}$ ${\frac {31711}{30}}$ $-{\frac {5462}{15}}$ ${\frac {32582}{45}}$ ${\frac {1025}{18}}$ ${\frac {2911}{30}}$

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=$2 is the signature of K11n160. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n159 K11n161