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L11n451 L11n453

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,20,12,21 X13,19,14,22 X21,18,22,9 X17,13,18,12 X8,16,5,15 X14,8,15,7 X19,17,20,16 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 6q-7/2 + 5q-5/2 - 7q-3/2 + 4q-1/2 - 6q1/2 + 2q3/2 - q5/2 - q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + 2q-14 + 7q-12 + 7q-10 + 10q-8 + 12q-6 + 10q-4 + 12q-2 + 7 + 7q2 + 4q4 + q6 + 2q8 - q10 - q12 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 3a-3z + a-3z3 - a-1z-3 - 6a-1z-1 - 11a-1z - 7a-1z3 - a-1z5 + 3az-3 + 11az-1 + 15az + 9az3 + 2az5 - 3a3z-3 - 8a3z-1 - 8a3z - 3a3z3 + a5z-3 + 2a5z-1 + a5z

Kauffman Polynomial:

a-4 - 4a-4z2 + 5a-4z4 - a-4z6 - 2a-3z-1 + 9a-3z - 11a-3z3 + 7a-3z5 - a-3z7 + 6a-2 - 15a-2z2 + 7a-2z4 + a-1z-3 - 11a-1z-1 + 35a-1z - 51a-1z3 + 27a-1z5 - 4a-1z7 - 3z-2 + 18 - 23z2 - 7z4 + 17z6 - 4z8 + 3az-3 - 18az-1 + 50az - 75az3 + 44az5 - 5az7 - az9 - 6a2z-2 + 21a2 - 19a2z2 - 12a2z4 + 23a2z6 - 6a2z8 + 3a3z-3 - 14a3z-1 + 34a3z - 45a3z3 + 29a3z5 - 3a3z7 - a3z9 - 3a4z-2 + 9a4 - 7a4z2 - 3a4z4 + 7a4z6 - 2a4z8 + a5z-3 - 5a5z-1 + 10a5z - 10a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                         j = 6                 2 1 1   j = 4               2 1       j = 2             5 2 1       j = 0           5 8 1         j = -2         2 3 4           j = -4       3 5               j = -6     3 2                 j = -8   1 5                   j = -10   1                     j = -12 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]:=	Length[Skeleton[L]]
Out[2]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 452]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 452]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 20, 12, 21], X[13, 19, 14, 22], > X[21, 18, 22, 9], X[17, 13, 18, 12], X[8, 16, 5, 15], X[14, 8, 15, 7], > X[19, 17, 20, 16], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 8, -7}, {-9, 3, -5, 4}, > {11, -2, -3, 6, -4, -8, 7, 9, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 6 5 7 4 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 6 Sqrt[q] + 2 q - q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 9/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 2 7 7 10 12 10 12 2 4 6 7 + q + --- + --- + --- + --- + -- + -- + -- + -- + 7 q + 4 q + q + 16 14 12 10 8 6 4 2 q q q q q q q q 8 10 12 14 > 2 q - q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 452]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 1 6 11 a 8 a 2 a 3 z 11 z -(----) + --- - ---- + -- + ---- - --- + ---- - ---- + ---- + --- - ---- + 3 3 3 3 3 a z z z z 3 a a z z z z a z a 3 3 5 3 5 z 7 z 3 3 3 z 5 > 15 a z - 8 a z + a z + -- - ---- + 9 a z - 3 a z - -- + 2 a z 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 452]][a, z]
Out[9]=	3 5 2 4 -4 6 2 4 1 3 a 3 a a 3 6 a 3 a 18 + a + -- + 21 a + 9 a + ---- + --- + ---- + -- - -- - ---- - ---- - 2 3 3 3 3 2 2 2 a a z z z z z z z 3 5 2 11 18 a 14 a 5 a 9 z 35 z 3 > ---- - --- - ---- - ----- - ---- + --- + ---- + 50 a z + 34 a z + 3 a z z z z 3 a a z a 2 2 3 3 5 2 4 z 15 z 2 2 4 2 11 z 51 z > 10 a z - 23 z - ---- - ----- - 19 a z - 7 a z - ----- - ----- - 4 2 3 a a a a 4 4 3 3 3 5 3 4 5 z 7 z 2 4 4 4 > 75 a z - 45 a z - 10 a z - 7 z + ---- + ---- - 12 a z - 3 a z + 4 2 a a 5 5 6 7 z 27 z 5 3 5 5 5 6 z 2 6 > ---- + ----- + 44 a z + 29 a z + 5 a z + 17 z - -- + 23 a z + 3 a 4 a a 7 7 4 6 z 4 z 7 3 7 5 7 8 2 8 4 8 > 7 a z - -- - ---- - 5 a z - 3 a z - a z - 4 z - 6 a z - 2 a z - 3 a a 9 3 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 2 1 1 1 5 3 2 3 8 + -- + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 2 12 6 10 5 8 5 8 4 6 4 6 3 4 3 q q t q t q t q t q t q t q t 5 2 5 3 2 4 2 2 4 2 6 2 > ----- + ----- + - + ---- + t + 2 q t + 2 q t + q t + q t + 2 q t + 4 2 2 2 t 2 q t q t q t 6 3 6 4 10 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n452

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