K11n41

K11n40

K11n42

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n41 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n41/ThurstonBennequinNumber Hyperbolic Volume 13.7623 A-Polynomial See Data:K11n41/A-polynomial

[edit Notes for K11n41's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n41's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+8t^{2}-9t+9-9t^{-1}+8t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 53, 4 }
Jones polynomial	$q^{8}-4q^{7}+6q^{6}-8q^{5}+9q^{4}-8q^{3}+8q^{2}-5q+3-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-4z^{4}a^{-2}+13z^{4}a^{-4}-5z^{4}a^{-6}-4z^{2}a^{-2}+14z^{2}a^{-4}-8z^{2}a^{-6}+z^{2}a^{-8}-a^{-2}+6a^{-4}-5a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+9z^{8}a^{-4}+6z^{8}a^{-6}+z^{7}a^{-1}-3z^{7}a^{-3}+2z^{7}a^{-5}+6z^{7}a^{-7}-13z^{6}a^{-2}-36z^{6}a^{-4}-21z^{6}a^{-6}+2z^{6}a^{-8}-4z^{5}a^{-1}-12z^{5}a^{-3}-28z^{5}a^{-5}-20z^{5}a^{-7}+17z^{4}a^{-2}+42z^{4}a^{-4}+24z^{4}a^{-6}-z^{4}a^{-8}+5z^{3}a^{-1}+20z^{3}a^{-3}+35z^{3}a^{-5}+24z^{3}a^{-7}+4z^{3}a^{-9}-8z^{2}a^{-2}-22z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-10}-2za^{-1}-7za^{-3}-13za^{-5}-10za^{-7}-2za^{-9}+a^{-2}+6a^{-4}+5a^{-6}+a^{-8}$
The A2 invariant	$-q^{2}+1-q^{-2}+q^{-4}+q^{-6}+q^{-8}+4q^{-10}-q^{-12}+3q^{-14}-2q^{-16}-q^{-18}-q^{-20}-2q^{-22}+q^{-24}-q^{-26}+q^{-28}$
The G2 invariant	Data:K11n41/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n41 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["K11n41"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-4t^{3}+8t^{2}-9t+9-9t^{-1}+8t^{-2}-4t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 53, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-4z^{4}a^{-2}+13z^{4}a^{-4}-5z^{4}a^{-6}-4z^{2}a^{-2}+14z^{2}a^{-4}-8z^{2}a^{-6}+z^{2}a^{-8}-a^{-2}+6a^{-4}-5a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+9z^{8}a^{-4}+6z^{8}a^{-6}+z^{7}a^{-1}-3z^{7}a^{-3}+2z^{7}a^{-5}+6z^{7}a^{-7}-13z^{6}a^{-2}-36z^{6}a^{-4}-21z^{6}a^{-6}+2z^{6}a^{-8}-4z^{5}a^{-1}-12z^{5}a^{-3}-28z^{5}a^{-5}-20z^{5}a^{-7}+17z^{4}a^{-2}+42z^{4}a^{-4}+24z^{4}a^{-6}-z^{4}a^{-8}+5z^{3}a^{-1}+20z^{3}a^{-3}+35z^{3}a^{-5}+24z^{3}a^{-7}+4z^{3}a^{-9}-8z^{2}a^{-2}-22z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-10}-2za^{-1}-7za^{-3}-13za^{-5}-10za^{-7}-2za^{-9}+a^{-2}+6a^{-4}+5a^{-6}+a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n47,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n47,}

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["K11n41"];

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]=

{ $t^{4}-4t^{3}+8t^{2}-9t+9-9t^{-1}+8t^{-2}-4t^{-3}+t^{-4}$, $q^{8}-4q^{7}+6q^{6}-8q^{5}+9q^{4}-8q^{3}+8q^{2}-5q+3-q^{-1}$ }

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]=

{K11n47,}

In[6]:=

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

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

Out[6]=

{K11n47,}

Vassiliev invariants

V2 and V3:

(3, 4)

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 $12$ $32$ $72$ $110$ $2$ $384$ ${\frac {1472}{3}}$ ${\frac {128}{3}}$ $64$ $288$ $512$ $1320$ $24$ ${\frac {21471}{10}}$ ${\frac {614}{15}}$ ${\frac {3194}{5}}$ ${\frac {43}{2}}$ ${\frac {511}{10}}$

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=$4 is the signature of K11n41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11n40 K11n42