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L11n438 L11n440

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DrawMorseLink

PD Presentation:

X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X22,19,13,20 X20,15,21,16 X16,21,17,22 X17,9,18,12

Gauss Code:

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

Jones Polynomial:

q-9/2 - q-7/2 + q-5/2 + q-3/2 - 3q-1/2 + q1/2 - 5q3/2 + 2q5/2 - 5q7/2 + q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-14 - q-12 - 3q-10 - 3q-8 - 5q-6 - 4q-4 + 3 + 11q2 + 14q4 + 18q6 + 18q8 + 14q10 + 11q12 + 5q14 + 3q16 + q18

HOMFLY-PT Polynomial:

- 2a-5z-3 - 3a-5z-1 - a-5z + 7a-3z-3 + 14a-3z-1 + 11a-3z + 3a-3z3 - 9a-1z-3 - 22a-1z-1 - 22a-1z - 11a-1z3 - 2a-1z5 + 5az-3 + 14az-1 + 15az + 7az3 + az5 - a3z-3 - 3a3z-1 - 3a3z - a3z3

Kauffman Polynomial:

2a-5z-3 - 9a-5z-1 + 16a-5z - 14a-5z3 + 6a-5z5 - a-5z7 - 7a-4z-2 + 23a-4 - 26a-4z2 + 9a-4z4 + 2a-4z6 - a-4z8 + 7a-3z-3 - 23a-3z-1 + 39a-3z - 46a-3z3 + 30a-3z5 - 6a-3z7 - 19a-2z-2 + 60a-2 - 73a-2z2 + 37a-2z4 - 3a-2z6 - a-2z8 + 9a-1z-3 - 24a-1z-1 + 37a-1z - 46a-1z3 + 28a-1z5 - 5a-1z7 - 18z-2 + 58 - 75z2 + 40z4 - 6z6 + 5az-3 - 12az-1 + 16az - 18az3 + 9az5 - az7 - 7a2z-2 + 24a2 - 34a2z2 + 17a2z4 - 2a2z6 + a3z-3 - 2a3z-1 + 2a3z - 4a3z3 + 5a3z5 - a3z7 - a4z-2 + 4a4 - 6a4z2 + 5a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                         j = 8                   5 1   j = 6                 1 4     j = 4               4 1       j = 2           4 1 1         j = 0         1 7 4           j = -2       1   1             j = -4       1 3               j = -6   1 1                   j = -8                         j = -10 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, 439]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 439]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[11, 19, 12, 18], X[3, 11, 4, 10], > X[9, 1, 10, 4], X[7, 15, 8, 14], X[13, 5, 14, 8], X[22, 19, 13, 20], > X[20, 15, 21, 16], X[16, 21, 17, 22], X[17, 9, 18, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, > {-7, 6, 9, -10, -11, 3, 8, -9, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) -(7/2) -(5/2) -(3/2) 3 3/2 5/2 q - q + q + q - ------- + Sqrt[q] - 5 q + 2 q - Sqrt[q] 7/2 9/2 11/2 > 5 q + q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 3 3 5 4 2 4 6 8 10 3 - q - q - --- - -- - -- - -- + 11 q + 14 q + 18 q + 18 q + 14 q + 10 8 6 4 q q q q 12 14 16 18 > 11 q + 5 q + 3 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 439]][a, z]
Out[8]=	3 3 -2 7 9 5 a a 3 14 22 14 a 3 a z 11 z ----- + ----- - ---- + --- - -- - ---- + ---- - --- + ---- - ---- - -- + ---- - 5 3 3 3 3 3 3 5 3 a z z z 5 3 a z a z a z z z a z a z a a 3 3 5 22 z 3 3 z 11 z 3 3 3 2 z 5 > ---- + 15 a z - 3 a z + ---- - ----- + 7 a z - a z - ---- + a z a 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 439]][a, z]
Out[9]=	3 23 60 2 4 2 7 9 5 a a 18 7 58 + -- + -- + 24 a + 4 a + ----- + ----- + ---- + --- + -- - -- - ----- - 4 2 5 3 3 3 3 3 3 2 4 2 a a a z a z a z z z z a z 2 4 3 19 7 a a 9 23 24 12 a 2 a 16 z 39 z 37 z > ----- - ---- - -- - ---- - ---- - --- - ---- - ---- + ---- + ---- + ---- + 2 2 2 2 5 3 a z z z 5 3 a a z z z a z a z a a 2 2 3 3 2 26 z 73 z 2 2 4 2 14 z > 16 a z + 2 a z - 75 z - ----- - ----- - 34 a z - 6 a z - ----- - 4 2 5 a a a 3 3 4 4 46 z 46 z 3 3 3 4 9 z 37 z 2 4 > ----- - ----- - 18 a z - 4 a z + 40 z + ---- + ----- + 17 a z + 3 a 4 2 a a a 5 5 5 6 6 4 4 6 z 30 z 28 z 5 3 5 6 2 z 3 z > 5 a z + ---- + ----- + ----- + 9 a z + 5 a z - 6 z + ---- - ---- - 5 3 a 4 2 a a a a 7 7 7 8 8 2 6 4 6 z 6 z 5 z 7 3 7 z z > 2 a z - a z - -- - ---- - ---- - a z - a z - -- - -- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 1 1 1 3 7 + q + 4 q + ------ + ----- + ----- + ----- + ----- + - + ---- + 4 t + 10 5 6 4 6 3 4 2 2 2 t 4 q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 6 4 8 4 8 5 12 6 > q t + q t + 4 q t + q t + q t + 4 q t + 5 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n439

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