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L11n450 L11n452

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X20,12,21,11 X22,13,19,14 X18,22,9,21 X12,17,13,18 X15,8,16,5 X7,14,8,15 X16,19,17,20 X2536 X9,1,10,4

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-15/2 - 3q-13/2 + 5q-11/2 - 7q-9/2 + 5q-7/2 - 7q-5/2 + 3q-3/2 - 4q-1/2 - q1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - q-18 + 3q-16 + 4q-14 + 6q-12 + 11q-10 + 10q-8 + 14q-6 + 10q-4 + 9q-2 + 8 + 3q2 + 3q4 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-3 - 3a-1z-1 - 3a-1z - a-1z3 + 3az-3 + 9az-1 + 12az + 7az3 + az5 - 3a3z-3 - 10a3z-1 - 15a3z - 9a3z3 - 2a3z5 + a5z-3 + 5a5z-1 + 7a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-1z-3 - 5a-1z-1 + 10a-1z - 10a-1z3 + 6a-1z5 - a-1z7 - 3z-2 + 9 - 6z2 - 5z4 + 6z6 - z8 + 3az-3 - 14az-1 + 38az - 49az3 + 20az5 - 2az7 - 6a2z-2 + 21a2 - 27a2z2 + 2a2z4 + 5a2z6 - a2z8 + 3a3z-3 - 18a3z-1 + 54a3z - 73a3z3 + 39a3z5 - 7a3z7 - 3a4z-2 + 18a4 - 33a4z2 + 19a4z4 + a4z6 - 2a4z8 + a5z-3 - 11a5z-1 + 31a5z - 43a5z3 + 35a5z5 - 9a5z7 + 6a6 - 15a6z2 + 15a6z4 + a6z6 - 2a6z8 - 2a7z-1 + 5a7z - 9a7z3 + 10a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                       1 j = 4                         j = 2                   1 1   j = 0               6 1       j = -2             4 6 1       j = -4           4 2 2         j = -6         2 4             j = -8       5 4 1             j = -10     2 4                 j = -12   1 3                   j = -14   2                     j = -16 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, 451]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 451]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[20, 12, 21, 11], X[22, 13, 19, 14], > X[18, 22, 9, 21], X[12, 17, 13, 18], X[15, 8, 16, 5], X[7, 14, 8, 15], > X[16, 19, 17, 20], X[2, 5, 3, 6], X[9, 1, 10, 4]]
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]=	-(15/2) 3 5 7 5 7 3 4 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- - Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 -18 3 4 6 11 10 14 10 9 8 - q - q + q - q + --- + --- + --- + --- + -- + -- + -- + -- + 16 14 12 10 8 6 4 2 q q q q q q q q 2 4 6 8 > 3 q + 3 q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 451]][a, z]
Out[8]=	3 5 3 5 7 1 3 a 3 a a 3 9 a 10 a 5 a a 3 z -(----) + --- - ---- + -- - --- + --- - ----- + ---- - -- - --- + 12 a z - 3 3 3 3 a z z z z z a a z z z z 3 3 5 7 z 3 3 3 5 3 5 3 5 > 15 a z + 7 a z - a z - -- + 7 a z - 9 a z + 3 a z + a z - 2 a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 451]][a, z]
Out[9]=	3 5 2 4 2 4 6 8 1 3 a 3 a a 3 6 a 3 a 9 + 21 a + 18 a + 6 a + a + ---- + --- + ---- + -- - -- - ---- - ---- - 3 3 3 3 2 2 2 a z z z z z z z 3 5 7 5 14 a 18 a 11 a 2 a 10 z 3 5 > --- - ---- - ----- - ----- - ---- + ---- + 38 a z + 54 a z + 31 a z + a z z z z z a 3 7 2 2 2 4 2 6 2 8 2 10 z > 5 a z - 6 z - 27 a z - 33 a z - 15 a z - 3 a z - ----- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 49 a z - 73 a z - 43 a z - 9 a z - 5 z + 2 a z + 19 a z + 5 6 4 8 4 6 z 5 3 5 5 5 7 5 > 15 a z + 3 a z + ---- + 20 a z + 39 a z + 35 a z + 10 a z + a 7 6 2 6 4 6 6 6 8 6 z 7 3 7 5 7 > 6 z + 5 a z + a z + a z - a z - -- - 2 a z - 7 a z - 9 a z - a 7 7 8 2 8 4 8 6 8 > 3 a z - z - a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 6 1 2 1 3 2 4 5 6 + -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 4 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 q q q t q t q t q t q t q t q t 4 2 1 4 4 2 4 t 2 2 > ----- + ----- + ----- + ----- + ----- + ---- + ---- + t + -- + q t + 8 3 6 3 8 2 6 2 4 2 4 2 2 q t q t q t q t q t q t q t q 2 3 6 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n451

L11n450 L11n452