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L11n350 L11n352

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DrawMorseLink

PD Presentation:

X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X9,13,10,22 X21,17,22,16 X17,8,18,9 X15,11,16,10 X19,12,20,5 X2536 X13,1,14,4

Gauss Code:

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

Jones Polynomial:

3q-4 - 6q-3 + 11q-2 - 13q-1 + 15 - 13q + 12q2 - 7q3 + 3q4 - q5

A2 (sl(3)) Invariant:

2q-14 + 4q-12 + 5q-8 + 3q-6 + q-4 + 6q-2 + 5q2 + q6 + 3q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + 2a-2 + 3a-2z2 + 2a-2z4 + z-2 + 2 - z2 - 2z4 - z6 - 2a2z-2 - 5a2 - 2a2z2 + a2z4 + a4z-2 + 2a4

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 + a-3z + 2a-3z3 - 7a-3z5 + 5a-3z7 - a-2z2 - 5a-2z6 + 5a-2z8 - 3a-1z + 12a-1z3 - 18a-1z5 + 7a-1z7 + 2a-1z9 - z-2 + 9 - 22z2 + 24z4 - 20z6 + 10z8 + 2az-1 - 8az + 12az3 - 13az5 + 5az7 + 2az9 - 2a2z-2 + 13a2 - 29a2z2 + 25a2z4 - 12a2z6 + 5a2z8 + 2a3z-1 - 5a3z + 4a3z3 - 3a3z5 + 3a3z7 - a4z-2 + 6a4 - 11a4z2 + 6a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                   1 j = 9                 2   j = 7               5 1   j = 5             7 2     j = 3           6 5       j = 1         9 7         j = -1       7 9           j = -3     4 6             j = -5   2 7               j = -7 1 4                 j = -9 3

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]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 351]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 351]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 15, 4, 14], X[11, 20, 12, 21], X[7, 18, 8, 19], > X[9, 13, 10, 22], X[21, 17, 22, 16], X[17, 8, 18, 9], X[15, 11, 16, 10], > X[19, 12, 20, 5], X[2, 5, 3, 6], X[13, 1, 14, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -4, 7, -5, 8, -3, 9}, > {-11, 2, -8, 6, -7, 4, -9, 3, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	3 6 11 13 2 3 4 5 15 + -- - -- + -- - -- - 13 q + 12 q - 7 q + 3 q - q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 5 3 -4 6 2 6 8 10 12 16 --- + --- + -- + -- + q + -- + 5 q + q + 3 q - 3 q + q - q 14 12 8 6 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 351]][a, z]
Out[8]=	2 4 2 2 -4 2 2 4 -2 2 a a 2 z 3 z 2 2 2 - a + -- - 5 a + 2 a + z - ---- + -- - z - -- + ---- - 2 a z - 2 2 2 4 2 a z z a a 4 4 2 z 2 4 6 > 2 z + ---- + a z - z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 351]][a, z]
Out[9]=	2 4 3 -4 2 4 -2 2 a a 2 a 2 a z z 3 z 9 - a + 13 a + 6 a - z - ---- - -- + --- + ---- + -- + -- - --- - 8 a z - 2 2 z z 5 3 a z z a a 2 2 3 3 3 3 2 3 z z 2 2 4 2 2 z 2 z 12 z > 5 a z - 22 z + ---- - -- - 29 a z - 11 a z - ---- + ---- + ----- + 4 2 5 3 a a a a a 4 5 5 5 3 3 3 4 5 z 2 4 4 4 z 7 z 18 z > 12 a z + 4 a z + 24 z - ---- + 25 a z + 6 a z + -- - ---- - ----- - 4 5 3 a a a a 6 6 7 7 5 3 5 6 3 z 5 z 2 6 5 z 7 z 7 > 13 a z - 3 a z - 20 z + ---- - ---- - 12 a z + ---- + ---- + 5 a z + 4 2 3 a a a a 8 9 3 7 8 5 z 2 8 2 z 9 > 3 a z + 10 z + ---- + 5 a z + ---- + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	9 3 1 4 2 7 4 6 7 - + 9 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + q 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 6 q t + 5 q t + 7 q t + 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n351

L11n350 L11n352