© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n107 K11n109

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,14,6,15 X7,16,8,17 X18,9,19,10 X2,11,3,12 X20,13,21,14 X15,6,16,7 X22,18,1,17 X12,19,13,20 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 -14 -16 18 2 20 -6 22 12 8

Alexander Polynomial:

- t-3 + 8t-2 - 17t-1 + 21 - 17t + 8t2 - t3

Conway Polynomial:

1 + 6z2 + 2z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{73, -4}

Jones Polynomial:

- q-11 + 3q-10 - 7q-9 + 10q-8 - 12q-7 + 13q-6 - 11q-5 + 9q-4 - 5q-3 + 2q-2

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-34 + q-30 - 3q-28 + q-26 - q-24 - q-22 + 3q-20 - q-18 + 4q-16 - q-14 + 2q-10 - 2q-8 + 2q-6

HOMFLY-PT Polynomial:

a4 + 4a4z2 + 2a4z4 + a6 - 2a6z4 - a6z6 + 3a8z2 + 2a8z4 - a10 - a10z2

Kauffman Polynomial:

a4 - 4a4z2 + 3a4z4 - 4a5z3 + 3a5z5 + a5z7 - a6 + 2a6z2 - 3a6z4 + a6z6 + 2a6z8 + a7z + a7z3 - 7a7z5 + 5a7z7 + a7z9 + 3a8z2 - 3a8z4 - 5a8z6 + 6a8z8 - 3a9z + 13a9z3 - 20a9z5 + 9a9z7 + a9z9 + a10 - a10z2 - 2a10z4 - 3a10z6 + 4a10z8 - 3a11z + 6a11z3 - 9a11z5 + 5a11z7 + 2a12z2 - 5a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{6, -15}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-4 is the signature of 11108. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                   2 j = -5                 4 1 j = -7               5 1   j = -9             6 4     j = -11           7 5       j = -13         5 6         j = -15       5 7           j = -17     2 5             j = -19   1 5               j = -21   2                 j = -23 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]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 108]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 108]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 16, 8, 17], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 13, 21, 14], X[15, 6, 16, 7], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 108]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 8, -4, -11, 5, -2, 6, -10, 7, 3, -8, 4, 9, -5, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 108]]
Out[5]=	DTCode[4, 10, -14, -16, 18, 2, 20, -6, 22, 12, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 108]][t]
Out[6]=	-3 8 17 2 3 21 - t + -- - -- - 17 t + 8 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 108]][z]
Out[7]=	2 4 6 1 + 6 z + 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 108]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 108]], KnotSignature[Knot[11, NonAlternating, 108]]}
Out[9]=	{73, -4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 108]][q]
Out[10]=	-11 3 7 10 12 13 11 9 5 2 -q + --- - -- + -- - -- + -- - -- + -- - -- + -- 10 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 108]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 108]][q]
Out[12]=	-34 -30 3 -26 -24 -22 3 -18 4 -14 2 2 -q + q - --- + q - q - q + --- - q + --- - q + --- - -- + 28 20 16 10 8 q q q q q 2 > -- 6 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 108]][a, z]
Out[13]=	4 6 10 4 2 8 2 10 2 4 4 6 4 8 4 6 6 a + a - a + 4 a z + 3 a z - a z + 2 a z - 2 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 108]][a, z]
Out[14]=	4 6 10 7 9 11 13 4 2 6 2 8 2 a - a + a + a z - 3 a z - 3 a z + a z - 4 a z + 2 a z + 3 a z - 10 2 12 2 5 3 7 3 9 3 11 3 13 3 > a z + 2 a z - 4 a z + a z + 13 a z + 6 a z - 2 a z + 4 4 6 4 8 4 10 4 12 4 5 5 7 5 > 3 a z - 3 a z - 3 a z - 2 a z - 5 a z + 3 a z - 7 a z - 9 5 11 5 13 5 6 6 8 6 10 6 12 6 > 20 a z - 9 a z + a z + a z - 5 a z - 3 a z + 3 a z + 5 7 7 7 9 7 11 7 6 8 8 8 10 8 > a z + 5 a z + 9 a z + 5 a z + 2 a z + 6 a z + 4 a z + 7 9 9 9 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 108]], Vassiliev[3][Knot[11, NonAlternating, 108]]}
Out[15]=	{6, -15}
In[16]:=	Kh[Knot[11, NonAlternating, 108]][q, t]
Out[16]=	-5 2 1 2 1 5 2 5 5 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q t q t q t q t q t q t q t 7 5 6 7 5 6 4 5 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 4 > ---- 5 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n108

K11n107 K11n109