© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a238 K11a240

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X18,5,19,6 X20,8,21,7 X22,9,1,10 X6,12,7,11 X2,14,3,13 X8,15,9,16 X10,18,11,17 X12,19,13,20 X16,22,17,21

Gauss Code:

{1, -7, 2, -1, 3, -6, 4, -8, 5, -9, 6, -10, 7, -2, 8, -11, 9, -3, 10, -4, 11, -5}

DT (Dowker-Thistlethwaite) Code:

4 14 18 20 22 6 2 8 10 12 16

Alexander Polynomial:

- t-4 + 7t-3 - 22t-2 + 42t-1 - 51 + 42t - 22t2 + 7t3 - t4

Conway Polynomial:

1 + z2 - z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{195, 2}

Jones Polynomial:

q-3 - 5q-2 + 12q-1 - 20 + 28q - 31q2 + 32q3 - 28q4 + 20q5 - 12q6 + 5q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-8 - 3q-6 + 4q-4 - 3q-2 - 1 + 6q2 - 5q4 + 8q6 - 3q8 + q10 + q12 - 6q14 + 5q16 - 3q18 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 - a-4 + 3a-4z2 + 5a-4z4 + 2a-4z6 + 2a-2 - 2a-2z2 - 6a-2z4 - 4a-2z6 - a-2z8 + z2 + 2z4 + z6

Kauffman Polynomial:

a-9z5 - 3a-8z4 + 5a-8z6 + 6a-7z3 - 14a-7z5 + 12a-7z7 - 3a-6z2 + 13a-6z4 - 24a-6z6 + 17a-6z8 + a-5z + 12a-5z3 - 21a-5z5 - 4a-5z7 + 13a-5z9 - a-4 - 8a-4z2 + 45a-4z4 - 72a-4z6 + 29a-4z8 + 4a-4z10 + a-3z + 10a-3z3 - 6a-3z5 - 33a-3z7 + 24a-3z9 - 2a-2 - 8a-2z2 + 45a-2z4 - 66a-2z6 + 23a-2z8 + 4a-2z10 + 8a-1z3 - 8a-1z5 - 12a-1z7 + 11a-1z9 - 3z2 + 15z4 - 22z6 + 11z8 + 4az3 - 8az5 + 5az7 - a2z4 + a2z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{1, 1}

Khovanov Homology: (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 11239. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     4   j = 13                   8 1   j = 11                 12 4     j = 9               16 8       j = 7             16 12         j = 5           15 16           j = 3         13 16             j = 1       8 16               j = -1     4 12                 j = -3   1 8                   j = -5   4                     j = -7 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 239]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 239]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[18, 5, 19, 6], X[20, 8, 21, 7], > X[22, 9, 1, 10], X[6, 12, 7, 11], X[2, 14, 3, 13], X[8, 15, 9, 16], > X[10, 18, 11, 17], X[12, 19, 13, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 239]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -6, 4, -8, 5, -9, 6, -10, 7, -2, 8, -11, 9, -3, 10, > -4, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 239]]
Out[5]=	DTCode[4, 14, 18, 20, 22, 6, 2, 8, 10, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 239]][t]
Out[6]=	-4 7 22 42 2 3 4 -51 - t + -- - -- + -- + 42 t - 22 t + 7 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 239]][z]
Out[7]=	2 6 8 1 + z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 239]}
In[9]:=	{KnotDet[Knot[11, Alternating, 239]], KnotSignature[Knot[11, Alternating, 239]]}
Out[9]=	{195, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 239]][q]
Out[10]=	-3 5 12 2 3 4 5 6 7 8 -20 + q - -- + -- + 28 q - 31 q + 32 q - 28 q + 20 q - 12 q + 5 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 239]}
In[12]:=	A2Invariant[Knot[11, Alternating, 239]][q]
Out[12]=	-8 3 4 3 2 4 6 8 10 12 14 -1 + q - -- + -- - -- + 6 q - 5 q + 8 q - 3 q + q + q - 6 q + 6 4 2 q q q 16 18 22 24 > 5 q - 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 239]][a, z]
Out[13]=	2 2 2 4 4 4 6 -4 2 2 z 3 z 2 z 4 z 5 z 6 z 6 2 z -a + -- + z - -- + ---- - ---- + 2 z - -- + ---- - ---- + z + ---- - 2 6 4 2 6 4 2 4 a a a a a a a a 6 8 4 z z > ---- - -- 2 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 239]][a, z]
Out[14]=	2 2 2 3 3 3 3 -4 2 z z 2 3 z 8 z 8 z 6 z 12 z 10 z 8 z -a - -- + -- + -- - 3 z - ---- - ---- - ---- + ---- + ----- + ----- + ---- + 2 5 3 6 4 2 7 5 3 a a a a a a a a a a 4 4 4 4 5 5 3 4 3 z 13 z 45 z 45 z 2 4 z 14 z > 4 a z + 15 z - ---- + ----- + ----- + ----- - a z + -- - ----- - 8 6 4 2 9 7 a a a a a a 5 5 5 6 6 6 6 21 z 6 z 8 z 5 6 5 z 24 z 72 z 66 z > ----- - ---- - ---- - 8 a z - 22 z + ---- - ----- - ----- - ----- + 5 3 a 8 6 4 2 a a a a a a 7 7 7 7 8 8 2 6 12 z 4 z 33 z 12 z 7 8 17 z 29 z > a z + ----- - ---- - ----- - ----- + 5 a z + 11 z + ----- + ----- + 7 5 3 a 6 4 a a a a a 8 9 9 9 10 10 23 z 13 z 24 z 11 z 4 z 4 z > ----- + ----- + ----- + ----- + ----- + ----- 2 5 3 a 4 2 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 239]], Vassiliev[3][Knot[11, Alternating, 239]]}
Out[15]=	{1, 1}
In[16]:=	Kh[Knot[11, Alternating, 239]][q, t]
Out[16]=	3 1 4 1 8 4 12 8 q 3 16 q + 13 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 16 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 15 q t + 16 q t + 16 q t + 12 q t + 16 q t + 8 q t + 12 q t + 11 5 13 5 13 6 15 6 17 7 > 4 q t + 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a239

K11a238 K11a240