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

K11n105 K11n107

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X5,15,6,14 X7,16,8,17 X2,10,3,9 X20,11,21,12 X22,13,1,14 X15,18,16,19 X17,8,18,9 X19,7,20,6 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 3t-2 + 6t-1 - 7 + 6t - 3t2 + t3

Conway Polynomial:

1 + 3z2 + 3z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{810, 10143, ...}

Determinant and Signature:

{27, -2}

Jones Polynomial:

- 2q-4 + 3q-3 - 3q-2 + 5q-1 - 4 + 4q - 3q2 + 2q3 - q4

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

{...}

A2 (sl(3)) Invariant:

- q-14 - 2q-12 + 2q-6 + 2q-4 + q-2 + 2 + q4 - q8 - q12

HOMFLY-PT Polynomial:

- 2a-2 - 3a-2z2 - a-2z4 + 4 + 8z2 + 5z4 + z6 - 2a2z2 - a2z4 - a4

Kauffman Polynomial:

- 3a-3z + 7a-3z3 - 5a-3z5 + a-3z7 + 2a-2 - 7a-2z2 + 14a-2z4 - 10a-2z6 + 2a-2z8 - 6a-1z + 14a-1z3 - 7a-1z5 - 2a-1z7 + a-1z9 + 4 - 15z2 + 26z4 - 19z6 + 4z8 - 4az + 9az3 - 5az5 - 2az7 + az9 - 8a2z2 + 13a2z4 - 9a2z6 + 2a2z8 + a3z + 2a3z3 - 3a3z5 + a3z7 - a4 + a4z4 + 2a5z

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

{3, -1}

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=-2 is the signature of 11106. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                 1 j = 7               1   j = 5             2 1   j = 3           2 1     j = 1         2 2       j = -1       3 2         j = -3     1 3           j = -5   2 2             j = -7   1               j = -9 2

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, 106]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 106]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 15, 6, 14], X[7, 16, 8, 17], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[22, 13, 1, 14], X[15, 18, 16, 19], > X[17, 8, 18, 9], X[19, 7, 20, 6], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 106]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 10, -4, 9, 5, -2, 6, -11, 7, 3, -8, 4, -9, 8, -10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, NonAlternating, 106]]
Out[5]=	DTCode[4, 10, -14, -16, 2, 20, 22, -18, -8, -6, 12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 106]][t]
Out[6]=	-3 3 6 2 3 -7 + t - -- + - + 6 t - 3 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 106]][z]
Out[7]=	2 4 6 1 + 3 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 10], Knot[10, 143], Knot[11, NonAlternating, 106]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 106]], KnotSignature[Knot[11, NonAlternating, 106]]}
Out[9]=	{27, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 106]][q]
Out[10]=	2 3 3 5 2 3 4 -4 - -- + -- - -- + - + 4 q - 3 q + 2 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 106]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 106]][q]
Out[12]=	-14 2 2 2 -2 4 8 12 2 - q - --- + -- + -- + q + q - q - q 12 6 4 q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 106]][a, z]
Out[13]=	2 4 2 4 2 3 z 2 2 4 z 2 4 6 4 - -- - a + 8 z - ---- - 2 a z + 5 z - -- - a z + z 2 2 2 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 106]][a, z]
Out[14]=	2 2 4 3 z 6 z 3 5 2 7 z 2 2 4 + -- - a - --- - --- - 4 a z + a z + 2 a z - 15 z - ---- - 8 a z + 2 3 a 2 a a a 3 3 4 5 7 z 14 z 3 3 3 4 14 z 2 4 4 4 5 z > ---- + ----- + 9 a z + 2 a z + 26 z + ----- + 13 a z + a z - ---- - 3 a 2 3 a a a 5 6 7 7 7 z 5 3 5 6 10 z 2 6 z 2 z 7 > ---- - 5 a z - 3 a z - 19 z - ----- - 9 a z + -- - ---- - 2 a z + a 2 3 a a a 8 9 3 7 8 2 z 2 8 z 9 > a z + 4 z + ---- + 2 a z + -- + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 106]], Vassiliev[3][Knot[11, NonAlternating, 106]]}
Out[15]=	{3, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 106]][q, t]
Out[16]=	3 3 2 1 2 2 1 2 t 2 3 2 -- + - + ----- + ----- + ----- + ---- + ---- + --- + 2 q t + 2 q t + 2 q t + 3 q 9 3 7 2 5 2 5 3 q q q t q t q t q t q t 3 3 5 3 5 4 7 4 9 5 > q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n106

K11n105 K11n107