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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X13,2,14,3

Gauss Code:

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

Jones Polynomial:

- q-9 + q-8 - 2q-7 + 4q-6 - 2q-5 + 3q-4 - q-3 + 2q-2

A2 (sl(3)) Invariant:

- q-32 - q-30 - 2q-28 - 2q-26 - q-24 + 4q-20 + 5q-18 + 7q-16 + 6q-14 + 4q-12 + 4q-10 + 2q-8 + 2q-6

HOMFLY-PT Polynomial:

2a4z-2 + 7a4 + 8a4z2 + 2a4z4 - 5a6z-2 - 13a6 - 13a6z2 - 6a6z4 - a6z6 + 4a8z-2 + 7a8 + 5a8z2 + a8z4 - a10z-2 - a10

Kauffman Polynomial:

- 2a4z-2 + 8a4 - 11a4z2 + 3a4z4 + 5a5z-1 - 13a5z + 11a5z3 - 5a5z5 + a5z7 - 5a6z-2 + 16a6 - 22a6z2 + 16a6z4 - 6a6z6 + a6z8 + 9a7z-1 - 24a7z + 25a7z3 - 11a7z5 + 2a7z7 - 4a8z-2 + 11a8 - 13a8z2 + 14a8z4 - 6a8z6 + a8z8 + 5a9z-1 - 14a9z + 15a9z3 - 6a9z5 + a9z7 - a10z-2 + 2a10 - 2a10z2 + a10z4 + a11z-1 - 3a11z + a11z3

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3               2 j = -5             1 2 j = -7           2     j = -9       1 1 1     j = -11       4 2       j = -13       2         j = -15   1 2           j = -17                 j = -19 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 376]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 376]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[22, 18, 19, 17], X[11, 20, 12, 21], X[19, 12, 20, 13], > X[18, 22, 5, 21], X[9, 16, 10, 17], X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-8, 7, 9, -6}, > {-4, -1, 2, 5, -10, 4, -7, 8, -11, -2, 3, 10, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-9 -8 2 4 2 3 -3 2 -q + q - -- + -- - -- + -- - q + -- 7 6 5 4 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 2 -24 4 5 7 6 4 4 2 2 -q - q - --- - --- - q + --- + --- + --- + --- + --- + --- + -- + -- 28 26 20 18 16 14 12 10 8 6 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 376]][a, z]
Out[8]=	4 6 8 10 4 6 8 10 2 a 5 a 4 a a 4 2 6 2 7 a - 13 a + 7 a - a + ---- - ---- + ---- - --- + 8 a z - 13 a z + 2 2 2 2 z z z z 8 2 4 4 6 4 8 4 6 6 > 5 a z + 2 a z - 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 376]][a, z]
Out[9]=	4 6 8 10 5 7 9 4 6 8 10 2 a 5 a 4 a a 5 a 9 a 5 a 8 a + 16 a + 11 a + 2 a - ---- - ---- - ---- - --- + ---- + ---- + ---- + 2 2 2 2 z z z z z z z 11 a 5 7 9 11 4 2 6 2 > --- - 13 a z - 24 a z - 14 a z - 3 a z - 11 a z - 22 a z - z 8 2 10 2 5 3 7 3 9 3 11 3 4 4 > 13 a z - 2 a z + 11 a z + 25 a z + 15 a z + a z + 3 a z + 6 4 8 4 10 4 5 5 7 5 9 5 6 6 > 16 a z + 14 a z + a z - 5 a z - 11 a z - 6 a z - 6 a z - 8 6 5 7 7 7 9 7 6 8 8 8 > 6 a z + a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 2 4 1 2 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 5 3 19 7 15 6 15 5 13 4 11 4 9 4 11 3 9 3 q q q t q t q t q t q t q t q t q t 1 2 1 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n376

L11n375 L11n377