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L10a173 L10n1

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DrawMorseLink

PD Presentation:

X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X20,15,17,16 X16,19,13,20 X12,17,9,18

Gauss Code:

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

Jones Polynomial:

q-12 - q-11 + 6q-10 - 6q-9 + 15q-8 - 11q-7 + 15q-6 - 10q-5 + 10q-4 - 4q-3 + q-2

A2 (sl(3)) Invariant:

q-40 + 5q-38 + 10q-36 + 15q-34 + 25q-32 + 31q-30 + 30q-28 + 35q-26 + 29q-24 + 23q-22 + 18q-20 + 8q-18 + 9q-16 + q-14 + 2q-12 + 3q-10 - 3q-8 + q-6

HOMFLY-PT Polynomial:

a4z4 + a6z-4 + 5a6z-2 + 10a6 + 10a6z2 + 4a6z4 - 4a8z-4 - 15a8z-2 - 20a8 - 10a8z2 + 6a10z-4 + 15a10z-2 + 10a10 - 4a12z-4 - 5a12z-2 + a14z-4

Kauffman Polynomial:

a4z4 + 4a5z5 - a6z-4 + 5a6z-2 - 10a6 + 10a6z2 - 14a6z4 + 10a6z6 + 4a7z-3 - 15a7z-1 + 20a7z - 10a7z3 - 10a7z5 + 10a7z7 - 4a8z-4 + 14a8z-2 - 25a8 + 30a8z2 - 25a8z4 + 5a8z6 + 5a8z8 + 12a9z-3 - 41a9z-1 + 55a9z - 30a9z3 - 12a9z5 + 11a9z7 + a9z9 - 6a10z-4 + 18a10z-2 - 31a10 + 40a10z2 - 25a10z4 - 2a10z6 + 6a10z8 + 12a11z-3 - 41a11z-1 + 55a11z - 30a11z3 + 2a11z5 + 2a11z7 + a11z9 - 4a12z-4 + 14a12z-2 - 25a12 + 30a12z2 - 20a12z4 + 4a12z6 + a12z8 + 4a13z-3 - 15a13z-1 + 20a13z - 10a13z3 + a13z7 - a14z-4 + 5a14z-2 - 10a14 + 10a14z2 - 5a14z4 + a14z6

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                     1 j = -5                   4 1 j = -7                 6     j = -9               4 4     j = -11             11 6       j = -13           10 14         j = -15         5 1           j = -17       1 10             j = -19     5 5               j = -21   1 6                 j = -23                       j = -25 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]=	5
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 174]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 174]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 11, 19, 12], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[14, 7, 15, 8], X[8, 13, 5, 14], X[20, 15, 17, 16], > X[16, 19, 13, 20], X[12, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10}, {7, -6, 8, -9}, > {10, -3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-12 -11 6 6 15 11 15 10 10 4 -2 q - q + --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 5 10 15 25 31 30 35 29 23 18 8 9 q + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + 38 36 34 32 30 28 26 24 22 20 18 16 q q q q q q q q q q q q -14 2 3 3 -6 > q + --- + --- - -- + q 12 10 8 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 174]][a, z]
Out[8]=	6 8 10 12 14 6 8 6 8 10 a 4 a 6 a 4 a a 5 a 15 a 10 a - 20 a + 10 a + -- - ---- + ----- - ----- + --- + ---- - ----- + 4 4 4 4 4 2 2 z z z z z z z 10 12 15 a 5 a 6 2 8 2 4 4 6 4 > ------ - ----- + 10 a z - 10 a z + a z + 4 a z 2 2 z z
In[9]:=	Kauffman[Link[10, Alternating, 174]][a, z]
Out[9]=	6 8 10 12 14 6 8 10 12 14 a 4 a 6 a 4 a a -10 a - 25 a - 31 a - 25 a - 10 a - -- - ---- - ----- - ----- - --- + 4 4 4 4 4 z z z z z 7 9 11 13 6 8 10 12 14 4 a 12 a 12 a 4 a 5 a 14 a 18 a 14 a 5 a > ---- + ----- + ------ + ----- + ---- + ----- + ------ + ------ + ----- - 3 3 3 3 2 2 2 2 2 z z z z z z z z z 7 9 11 13 15 a 41 a 41 a 15 a 7 9 11 13 > ----- - ----- - ------ - ------ + 20 a z + 55 a z + 55 a z + 20 a z + z z z z 6 2 8 2 10 2 12 2 14 2 7 3 > 10 a z + 30 a z + 40 a z + 30 a z + 10 a z - 10 a z - 9 3 11 3 13 3 4 4 6 4 8 4 > 30 a z - 30 a z - 10 a z + a z - 14 a z - 25 a z - 10 4 12 4 14 4 5 5 7 5 9 5 > 25 a z - 20 a z - 5 a z + 4 a z - 10 a z - 12 a z + 11 5 6 6 8 6 10 6 12 6 14 6 7 7 > 2 a z + 10 a z + 5 a z - 2 a z + 4 a z + a z + 10 a z + 9 7 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 11 a z + 2 a z + a z + 5 a z + 6 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 1 6 5 5 1 10 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 21 9 21 8 19 8 19 7 17 7 17 6 q t q t q t q t q t q t q t 5 1 10 14 11 6 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 15 6 15 5 13 5 13 4 11 4 11 3 9 3 9 2 q t q t q t q t q t q t q t q t 6 4 > ----- + ---- 7 2 5 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a174

L10a173 L10n1