K11n60

K11n59

K11n61

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n60 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,15,6,14 X2837 X9,17,10,16 X11,18,12,19 X13,20,14,21 X15,7,16,6 X17,1,18,22 X19,12,20,13 X21,10,22,11
Gauss code	1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 10, -7, 3, -8, 5, -9, 6, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 8 -14 2 -16 -18 -20 -6 -22 -12 -10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n60/ThurstonBennequinNumber Hyperbolic Volume 9.48716 A-Polynomial See Data:K11n60/A-polynomial

[edit Notes for K11n60's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n60's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-4t^{2}+5t-5+5t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-5z^{6}-6z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 31, 2 }
Jones polynomial	$-q^{6}+2q^{5}-3q^{4}+5q^{3}-5q^{2}+5q-4+3q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-7z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-17z^{4}a^{-2}+6z^{4}a^{-4}+5z^{4}-17z^{2}a^{-2}+11z^{2}a^{-4}-z^{2}a^{-6}+7z^{2}-6a^{-2}+6a^{-4}-2a^{-6}+3$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+2az^{7}-2z^{7}a^{-1}-3z^{7}a^{-3}+z^{7}a^{-5}+a^{2}z^{6}-20z^{6}a^{-2}-11z^{6}a^{-4}-8z^{6}-8az^{5}-4z^{5}a^{-1}-4z^{5}a^{-5}-4a^{2}z^{4}+35z^{4}a^{-2}+24z^{4}a^{-4}+2z^{4}a^{-6}+9z^{4}+7az^{3}+8z^{3}a^{-1}+8z^{3}a^{-3}+8z^{3}a^{-5}+z^{3}a^{-7}+3a^{2}z^{2}-25z^{2}a^{-2}-19z^{2}a^{-4}-4z^{2}a^{-6}-7z^{2}-2az-4za^{-1}-4za^{-3}-4za^{-5}-2za^{-7}+6a^{-2}+6a^{-4}+2a^{-6}+3$
The A2 invariant	$q^{8}+q^{4}-2q^{-4}+q^{-6}-q^{-8}+2q^{-10}+q^{-12}+q^{-14}+q^{-16}-q^{-18}-q^{-22}$
The G2 invariant	Data:K11n60/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n60 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["K11n60"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-5z^{6}-6z^{4}+1$

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]=

{ 31, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{6}+2q^{5}-3q^{4}+5q^{3}-5q^{2}+5q-4+3q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-7z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-17z^{4}a^{-2}+6z^{4}a^{-4}+5z^{4}-17z^{2}a^{-2}+11z^{2}a^{-4}-z^{2}a^{-6}+7z^{2}-6a^{-2}+6a^{-4}-2a^{-6}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+2az^{7}-2z^{7}a^{-1}-3z^{7}a^{-3}+z^{7}a^{-5}+a^{2}z^{6}-20z^{6}a^{-2}-11z^{6}a^{-4}-8z^{6}-8az^{5}-4z^{5}a^{-1}-4z^{5}a^{-5}-4a^{2}z^{4}+35z^{4}a^{-2}+24z^{4}a^{-4}+2z^{4}a^{-6}+9z^{4}+7az^{3}+8z^{3}a^{-1}+8z^{3}a^{-3}+8z^{3}a^{-5}+z^{3}a^{-7}+3a^{2}z^{2}-25z^{2}a^{-2}-19z^{2}a^{-4}-4z^{2}a^{-6}-7z^{2}-2az-4za^{-1}-4za^{-3}-4za^{-5}-2za^{-7}+6a^{-2}+6a^{-4}+2a^{-6}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_46,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_8,}

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["K11n60"];

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]=

{ $-t^{4}+3t^{3}-4t^{2}+5t-5+5t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$, $-q^{6}+2q^{5}-3q^{4}+5q^{3}-5q^{2}+5q-4+3q^{-1}-2q^{-2}+q^{-3}$ }

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]=

{10_46,}

In[6]:=

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

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

Out[6]=

{9_8,}

Vassiliev invariants

V2 and V3:

(0, 2)

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 $0$ $16$ $0$ $96$ $48$ $0$ ${\frac {544}{3}}$ ${\frac {160}{3}}$ $48$ $0$ $128$ $0$ $0$ $464$ $240$ $-112$ $112$ $-80$

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 K11n60. 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 χ 13                   1 -1 11                 1   1 9               2 1   -1 7             3 1     2 5           2 2       0 3         3 3         0 1       2 3           1 -1     1 2             -1 -3   1 2               1 -5   1                 -1 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n59 K11n61