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

K11a239 K11a241

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 18 20 22 16 2 10 12 6 8

Alexander Polynomial:

2t-4 - 5t-3 + 8t-2 - 10t-1 + 11 - 10t + 8t2 - 5t3 + 2t4

Conway Polynomial:

1 + 9z2 + 18z4 + 11z6 + 2z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{61, 8}

Jones Polynomial:

q4 - q5 + 4q6 - 5q7 + 7q8 - 9q9 + 9q10 - 9q11 + 7q12 - 5q13 + 3q14 - q15

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

{...}

A2 (sl(3)) Invariant:

q14 + 3q18 + q20 + 3q22 + q24 - q26 - 4q30 - 2q34 + q38 + q42 - q44

HOMFLY-PT Polynomial:

a-12 - 3a-12z2 - 4a-12z4 - a-12z6 - 6a-10 - 5a-10z2 + 5a-10z4 + 5a-10z6 + a-10z8 + 6a-8 + 17a-8z2 + 17a-8z4 + 7a-8z6 + a-8z8

Kauffman Polynomial:

a-19z3 - a-18z2 + 3a-18z4 + a-17z - 4a-17z3 + 5a-17z5 + a-16z2 - 7a-16z4 + 6a-16z6 + 2a-15z3 - 10a-15z5 + 6a-15z7 - a-14z2 + 4a-14z4 - 11a-14z6 + 5a-14z8 - a-13z + 5a-13z3 - 3a-13z5 - 6a-13z7 + 3a-13z9 + a-12 - a-12z2 + 12a-12z4 - 13a-12z6 + a-12z8 + a-12z10 - 7a-11z + 7a-11z3 + 13a-11z5 - 16a-11z7 + 4a-11z9 + 6a-10 - 15a-10z2 + 15a-10z4 - 3a-10z6 - 3a-10z8 + a-10z10 - 7a-9z + 9a-9z3 + a-9z5 - 4a-9z7 + a-9z9 + 6a-8 - 17a-8z2 + 17a-8z4 - 7a-8z6 + a-8z8

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

{9, 25}

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=8 is the signature of 11240. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 31                       1 j = 29                     2   j = 27                   3 1   j = 25                 4 2     j = 23               5 3       j = 21             4 4         j = 19           5 5           j = 17         2 4             j = 15       3 5               j = 13     1 2                 j = 11     3                   j = 9 1 1                     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, 240]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 240]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[20, 8, 21, 7], > X[22, 10, 1, 9], X[16, 12, 17, 11], X[2, 14, 3, 13], X[10, 16, 11, 15], > X[12, 18, 13, 17], X[6, 20, 7, 19], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 240]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -10, 4, -11, 5, -8, 6, -9, 7, -2, 8, -6, 9, -3, 10, > -4, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 240]]
Out[5]=	DTCode[4, 14, 18, 20, 22, 16, 2, 10, 12, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 240]][t]
Out[6]=	2 5 8 10 2 3 4 11 + -- - -- + -- - -- - 10 t + 8 t - 5 t + 2 t 4 3 2 t t t t
In[7]:=	Conway[Knot[11, Alternating, 240]][z]
Out[7]=	2 4 6 8 1 + 9 z + 18 z + 11 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 240]}
In[9]:=	{KnotDet[Knot[11, Alternating, 240]], KnotSignature[Knot[11, Alternating, 240]]}
Out[9]=	{61, 8}
In[10]:=	J=Jones[Knot[11, Alternating, 240]][q]
Out[10]=	4 5 6 7 8 9 10 11 12 13 14 q - q + 4 q - 5 q + 7 q - 9 q + 9 q - 9 q + 7 q - 5 q + 3 q - 15 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 240]}
In[12]:=	A2Invariant[Knot[11, Alternating, 240]][q]
Out[12]=	14 18 20 22 24 26 30 34 38 42 44 q + 3 q + q + 3 q + q - q - 4 q - 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 240]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -12 6 6 3 z 5 z 17 z 4 z 5 z 17 z z 5 z a - --- + -- - ---- - ---- + ----- - ---- + ---- + ----- - --- + ---- + 10 8 12 10 8 12 10 8 12 10 a a a a a a a a a a 6 8 8 7 z z z > ---- + --- + -- 8 10 8 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 240]][a, z]
Out[14]=	2 2 2 2 2 -12 6 6 z z 7 z 7 z z z z z 15 z a + --- + -- + --- - --- - --- - --- - --- + --- - --- - --- - ----- - 10 8 17 13 11 9 18 16 14 12 10 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 17 z z 4 z 2 z 5 z 7 z 9 z 3 z 7 z 4 z > ----- + --- - ---- + ---- + ---- + ---- + ---- + ---- - ---- + ---- + 8 19 17 15 13 11 9 18 16 14 a a a a a a a a a a 4 4 4 5 5 5 5 5 6 6 12 z 15 z 17 z 5 z 10 z 3 z 13 z z 6 z 11 z > ----- + ----- + ----- + ---- - ----- - ---- + ----- + -- + ---- - ----- - 12 10 8 17 15 13 11 9 16 14 a a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 8 13 z 3 z 7 z 6 z 6 z 16 z 4 z 5 z z 3 z z > ----- - ---- - ---- + ---- - ---- - ----- - ---- + ---- + --- - ---- + -- + 12 10 8 15 13 11 9 14 12 10 8 a a a a a a a a a a a 9 9 9 10 10 3 z 4 z z z z > ---- + ---- + -- + --- + --- 13 11 9 12 10 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 240]], Vassiliev[3][Knot[11, Alternating, 240]]}
Out[15]=	{9, 25}
In[16]:=	Kh[Knot[11, Alternating, 240]][q, t]
Out[16]=	7 9 9 11 2 13 2 13 3 15 3 15 4 q + q + q t + 3 q t + q t + 2 q t + 3 q t + 5 q t + 17 4 17 5 19 5 19 6 21 6 21 7 > 2 q t + 4 q t + 5 q t + 5 q t + 4 q t + 4 q t + 23 7 23 8 25 8 25 9 27 9 27 10 > 5 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 29 10 31 11 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a240

K11a239 K11a241