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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X18,11,19,12 X22,19,7,20 X20,14,21,13 X12,22,13,21 X14,17,15,18 X6718 X4,15,5,16

Gauss Code:

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

Jones Polynomial:

q-23/2 - 4q-21/2 + 7q-19/2 - 11q-17/2 + 14q-15/2 - 15q-13/2 + 14q-11/2 - 12q-9/2 + 8q-7/2 - 5q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 + 2q-32 + q-28 + 2q-26 - 2q-24 + 3q-22 - 2q-20 + q-18 + q-16 - 2q-14 + 3q-12 - q-10 + 2q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 4a3z3 - a3z5 + 2a5z-1 + 4a5z + 5a5z3 + 4a5z5 + a5z7 - 2a7z-1 - 4a7z + 3a7z5 + a7z7 + a9z-1 + a9z - 2a9z3 - a9z5

Kauffman Polynomial:

- a3z-1 + 5a3z - 8a3z3 + 5a3z5 - a3z7 + 3a4z2 - 9a4z4 + 8a4z6 - 2a4z8 - 2a5z-1 + 9a5z - 15a5z3 + 6a5z5 + 4a5z7 - 2a5z9 - a6 + 10a6z2 - 23a6z4 + 19a6z6 - 3a6z8 - a6z10 - 2a7z-1 + 9a7z - 12a7z3 - 2a7z5 + 14a7z7 - 6a7z9 + 6a8z2 - 22a8z4 + 26a8z6 - 8a8z8 - a8z10 - a9z-1 + 3a9z - 7a9z3 + 9a9z5 + a9z7 - 4a9z9 - 2a10z2 + 8a10z6 - 7a10z8 - 2a11z + a11z3 + 8a11z5 - 8a11z7 - a12z2 + 7a12z4 - 7a12z6 + 3a13z3 - 4a13z5 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                     1   j = -4                   4 1   j = -6                 5 2     j = -8               7 3       j = -10             7 5         j = -12           8 7           j = -14         6 7             j = -16       5 8               j = -18     3 7                 j = -20   1 4                   j = -22   3                     j = -24 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, 199]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 199]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 5, 17, 6], > X[18, 11, 19, 12], X[22, 19, 7, 20], X[20, 14, 21, 13], X[12, 22, 13, 21], > X[14, 17, 15, 18], X[6, 7, 1, 8], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, > {10, -1, 2, -3, 5, -8, 7, -9, 11, -4, 9, -5, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 4 7 11 14 15 14 12 8 5 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q 2 1 > ---- - ------- 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -28 2 2 3 2 -18 -16 2 3 -10 -q + --- + q + --- - --- + --- - --- + q + q - --- + --- - q + 32 26 24 22 20 14 12 q q q q q q q 2 -6 -2 > -- + q + q 8 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 199]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 9 3 3 -(--) + ---- - ---- + -- - 4 a z + 4 a z - 4 a z + a z - 4 a z + z z z z 5 3 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 5 a z - 2 a z - a z + 4 a z + 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 199]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 11 -a - -- - ---- - ---- - -- + 5 a z + 9 a z + 9 a z + 3 a z - 2 a z + z z z z 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 3 a z + 10 a z + 6 a z - 2 a z - a z - 8 a z - 15 a z - 7 3 9 3 11 3 13 3 4 4 6 4 8 4 > 12 a z - 7 a z + a z + 3 a z - 9 a z - 23 a z - 22 a z + 12 4 14 4 3 5 5 5 7 5 9 5 11 5 > 7 a z - a z + 5 a z + 6 a z - 2 a z + 9 a z + 8 a z - 13 5 4 6 6 6 8 6 10 6 12 6 3 7 > 4 a z + 8 a z + 19 a z + 26 a z + 8 a z - 7 a z - a z + 5 7 7 7 9 7 11 7 4 8 6 8 8 8 > 4 a z + 14 a z + a z - 8 a z - 2 a z - 3 a z - 8 a z - 10 8 5 9 7 9 9 9 6 10 8 10 > 7 a z - 2 a z - 6 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 3 1 4 3 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 8 6 7 8 7 7 5 7 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 3 5 t t 2 > ---- + ---- + -- + -- + t 8 6 4 2 q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a199

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