K11n42

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n42 at Knotilus!
	[edit K11n42 Quick Notes] 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.

[edit K11n42 Further Notes and Views]

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 X_12,5,13,6 X₂₈₃₇ X_9,18,10,19 X_11,21,12,20 X_6,13,7,14 X_15,10,16,11 X_17,22,18,1 X_19,15,20,14 X_21,16,22,17
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.

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.

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)

V_2,1 through V_6,9:

V_2,1	V_3,1	V_4,1	V_4,2	V_4,3	V_5,1	V_5,2	V_5,3	V_5,4	V_6,1	V_6,2	V_6,3	V_6,4	V_6,5	V_6,6	V_6,7	V_6,8	V_6,9
$0$	$16$	$0$	$-16$	$0$	$0$	$-{\frac {128}{3}}$	$-{\frac {128}{3}}$	$-16$	$0$	$128$	$0$	$0$	$312$	${\frac {32}{3}}$	$176$	$8$	$8$

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

5

1

0

3

1

2

1

0

1

2

1

2

-1

1

3

2

0

-3

2

1

-5

1

-1

-7

1

2

1

0

-9

1

0

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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