K11n42

K11n41

K11n43

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n42 at Knotilus!
	K11n42 is the mirror of the "Kinoshita-Terasaka" knot; it is a mutant of the (mirror of the) Conway knot K11n34. See also Heegaard Floer Knot Homology.

K11n42 is not $k$-colourable for any $k$. See The Determinant and the Signature.

Knot K11n42.

A graph, knot K11n42.

A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X9,18,10,19 X11,21,12,20 X6,13,7,14 X15,10,16,11 X17,22,18,1 X19,15,20,14 X21,16,22,17
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 5, -10, 6, -11, 9
Dowker-Thistlethwaite code	4 8 12 2 -18 -20 6 -10 -22 -14 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n42/ThurstonBennequinNumber Hyperbolic Volume 11.2191 A-Polynomial See Data:K11n42/A-polynomial

[edit Notes for K11n42's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n42's four dimensional invariants]

Polynomial invariants

Alexander polynomial	1
Conway polynomial	1
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 1, 0 }
Jones polynomial	$-q^{4}+2q^{3}-2q^{2}+2q+q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{6}+z^{6}+a^{4}z^{4}-6a^{2}z^{4}-z^{4}a^{-2}+6z^{4}+3a^{4}z^{2}-11a^{2}z^{2}-3z^{2}a^{-2}+11z^{2}+2a^{4}-6a^{2}-2a^{-2}+7$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+a^{4}z^{8}+2a^{2}z^{8}+2z^{8}a^{-2}+3z^{8}+2a^{5}z^{7}+2a^{3}z^{7}-5az^{7}-4z^{7}a^{-1}+z^{7}a^{-3}+a^{6}z^{6}-4a^{4}z^{6}-14a^{2}z^{6}-11z^{6}a^{-2}-20z^{6}-9a^{5}z^{5}-12a^{3}z^{5}-2z^{5}a^{-1}-5z^{5}a^{-3}-4a^{6}z^{4}+2a^{4}z^{4}+26a^{2}z^{4}+16z^{4}a^{-2}+36z^{4}+9a^{5}z^{3}+16a^{3}z^{3}+12az^{3}+11z^{3}a^{-1}+6z^{3}a^{-3}+3a^{6}z^{2}-2a^{4}z^{2}-20a^{2}z^{2}-9z^{2}a^{-2}-24z^{2}-3a^{5}z-7a^{3}z-7az-5za^{-1}-2za^{-3}+2a^{4}+6a^{2}+2a^{-2}+7$
The A2 invariant	$q^{18}+q^{14}-q^{12}-q^{10}-q^{8}-2q^{6}+q^{4}+3+2q^{-2}+q^{-4}+q^{-6}-q^{-8}-q^{-12}$
The G2 invariant	Data:K11n42/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n42 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["K11n42"];

In[4]:=

Alexander[K][t]

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

Out[4]=

1

In[5]:=

Conway[K][z]

Out[5]=

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]=

{ 1, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+a^{4}z^{8}+2a^{2}z^{8}+2z^{8}a^{-2}+3z^{8}+2a^{5}z^{7}+2a^{3}z^{7}-5az^{7}-4z^{7}a^{-1}+z^{7}a^{-3}+a^{6}z^{6}-4a^{4}z^{6}-14a^{2}z^{6}-11z^{6}a^{-2}-20z^{6}-9a^{5}z^{5}-12a^{3}z^{5}-2z^{5}a^{-1}-5z^{5}a^{-3}-4a^{6}z^{4}+2a^{4}z^{4}+26a^{2}z^{4}+16z^{4}a^{-2}+36z^{4}+9a^{5}z^{3}+16a^{3}z^{3}+12az^{3}+11z^{3}a^{-1}+6z^{3}a^{-3}+3a^{6}z^{2}-2a^{4}z^{2}-20a^{2}z^{2}-9z^{2}a^{-2}-24z^{2}-3a^{5}z-7a^{3}z-7az-5za^{-1}-2za^{-3}+2a^{4}+6a^{2}+2a^{-2}+7$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {0_1, K11n34,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n34,}

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["K11n42"];

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]=

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

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]=

{0_1, K11n34,}

In[6]:=

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

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

Out[6]=

{K11n34,}

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 $0$ $16$ $0$ $-16$ $0$ $0$ $-{\frac {128}{3}}$ $-{\frac {128}{3}}$ $-16$ $0$ $128$ $0$ $0$ $312$ ${\frac {32}{3}}$ $176$ $8$ $8$

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 K11n42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n41 K11n43