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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X20,15,21,16 X6,14,1,13 X18,11,19,12 X16,6,17,5 X12,17,13,18 X4,20,5,19 X14,21,15,22

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 4q-9/2 - 8q-7/2 + 14q-5/2 - 19q-3/2 + 21q-1/2 - 23q1/2 + 19q3/2 - 15q5/2 + 9q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 - 3q-14 + q-12 - q-10 - 3q-8 + 4q-6 - 2q-4 + 3q-2 + 3 + 6q4 - 3q6 + 4q8 + q10 - 3q12 + 2q14 - q16

HOMFLY-PT Polynomial:

2a-3z + 2a-3z3 + a-3z5 - 2a-1z-1 - 6a-1z - 8a-1z3 - 4a-1z5 - a-1z7 + 3az-1 + 8az + 8az3 + 3az5 - a3z-1 - 4a3z - 3a3z3 + a5z

Kauffman Polynomial:

- a-6z4 + a-5z3 - 4a-5z5 - 2a-4z2 + 7a-4z4 - 9a-4z6 + 4a-3z - 13a-3z3 + 20a-3z5 - 14a-3z7 - 6a-2z4 + 20a-2z6 - 14a-2z8 - 2a-1z-1 + 12a-1z - 35a-1z3 + 39a-1z5 - 3a-1z7 - 8a-1z9 + 3 + 6z2 - 43z4 + 61z6 - 20z8 - 2z10 - 3az-1 + 11az - 27az3 + 10az5 + 24az7 - 13az9 + 3a2 + 9a2z2 - 45a2z4 + 46a2z6 - 10a2z8 - 2a2z10 - a3z-1 + 4a3z - 9a3z3 - 2a3z5 + 12a3z7 - 5a3z9 + a4 + 5a4z2 - 16a4z4 + 14a4z6 - 4a4z8 + a5z - 3a5z3 + 3a5z5 - 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 = 12                       1 j = 10                     3   j = 8                   6 1   j = 6                 9 3     j = 4               11 7       j = 2             12 8         j = 0           10 12           j = -2         9 11             j = -4       5 10               j = -6     3 9                 j = -8   1 5                   j = -10   3                     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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 198]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 198]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[20, 15, 21, 16], X[6, 14, 1, 13], X[18, 11, 19, 12], X[16, 6, 17, 5], > X[12, 17, 13, 18], X[4, 20, 5, 19], X[14, 21, 15, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -10, 8, -6}, > {4, -1, 3, -2, 7, -9, 6, -11, 5, -8, 9, -7, 10, -5, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 8 14 19 21 3/2 -q + ---- - ---- + ---- - ---- + ------- - 23 Sqrt[q] + 19 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 -12 -10 3 4 2 3 4 6 8 10 3 + q - --- + q - q - -- + -- - -- + -- + 6 q - 3 q + 4 q + q - 14 8 6 4 2 q q q q q 12 14 16 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 198]][a, z]
Out[8]=	3 3 3 -2 3 a a 2 z 6 z 3 5 2 z 8 z 3 --- + --- - -- + --- - --- + 8 a z - 4 a z + a z + ---- - ---- + 8 a z - a z z z 3 a 3 a a a 5 5 7 3 3 z 4 z 5 z > 3 a z + -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 198]][a, z]
Out[9]=	3 2 4 2 3 a a 4 z 12 z 3 5 2 3 + 3 a + a - --- - --- - -- + --- + ---- + 11 a z + 4 a z + a z + 6 z - a z z z 3 a a 2 3 3 3 2 z 2 2 4 2 z 13 z 35 z 3 3 3 > ---- + 9 a z + 5 a z + -- - ----- - ----- - 27 a z - 9 a z - 4 5 3 a a a a 4 4 4 5 5 5 3 4 z 7 z 6 z 2 4 4 4 4 z 20 z > 3 a z - 43 z - -- + ---- - ---- - 45 a z - 16 a z - ---- + ----- + 6 4 2 5 3 a a a a a 5 6 6 39 z 5 3 5 5 5 6 9 z 20 z 2 6 > ----- + 10 a z - 2 a z + 3 a z + 61 z - ---- + ----- + 46 a z + a 4 2 a a 7 7 8 4 6 14 z 3 z 7 3 7 5 7 8 14 z > 14 a z - ----- - ---- + 24 a z + 12 a z - a z - 20 z - ----- - 3 a 2 a a 9 2 8 4 8 8 z 9 3 9 10 2 10 > 10 a z - 4 a z - ---- - 13 a z - 5 a z - 2 z - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 9 5 10 12 + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 9 10 11 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 8 q t + 11 q t + 7 q t + 9 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a198

L11a197 L11a199