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

K11n104 K11n106

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X11,21,12,20 X13,1,14,22 X18,16,19,15 X8,18,9,17 X6,19,7,20 X21,13,22,12

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 + 7t-2 - 16t-1 + 21 - 16t + 7t2 - t3

Conway Polynomial:

1 + 3z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{1078, K11n98, ...}

Determinant and Signature:

{69, 4}

Jones Polynomial:

2q2 - 4q3 + 8q4 - 10q5 + 12q6 - 12q7 + 9q8 - 7q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

2q6 - q8 + 3q10 + q12 - q14 + 3q16 - 2q18 + q20 - 2q22 - 2q24 + q26 - 2q28 + 2q30 + q32 - q34

HOMFLY-PT Polynomial:

- a-10z2 + 3a-8z2 + 2a-8z4 - 2a-6 - 4a-6z2 - 3a-6z4 - a-6z6 + 3a-4 + 5a-4z2 + 2a-4z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + a-12z2 - 7a-12z4 + 4a-12z6 + 5a-11z3 - 12a-11z5 + 6a-11z7 + a-10z2 - 4a-10z4 - 3a-10z6 + 4a-10z8 + a-9z + 11a-9z3 - 19a-9z5 + 8a-9z7 + a-9z9 - 2a-8z2 + 8a-8z4 - 10a-8z6 + 6a-8z8 - a-7z + 4a-7z3 - 5a-7z5 + 3a-7z7 + a-7z9 + 2a-6 - 8a-6z2 + 8a-6z4 - 3a-6z6 + 2a-6z8 - 2a-5z - a-5z3 + a-5z5 + a-5z7 + 3a-4 - 6a-4z2 + 3a-4z4

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

{3, 5}

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 11105. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                   1 j = 21                 3   j = 19               4 1   j = 17             5 3     j = 15           7 4       j = 13         5 5         j = 11       5 7           j = 9     3 5             j = 7   1 5               j = 5 1 3                 j = 3 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, 105]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 105]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[11, 21, 12, 20], X[13, 1, 14, 22], X[18, 16, 19, 15], > X[8, 18, 9, 17], X[6, 19, 7, 20], X[21, 13, 22, 12]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 105]]
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, 105]]
Out[5]=	DTCode[4, 10, 14, 16, 2, -20, -22, 18, 8, 6, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 105]][t]
Out[6]=	-3 7 16 2 3 21 - t + -- - -- - 16 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 105]][z]
Out[7]=	2 4 6 1 + 3 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 78], Knot[11, NonAlternating, 98], Knot[11, NonAlternating, 105]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 105]], KnotSignature[Knot[11, NonAlternating, 105]]}
Out[9]=	{69, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 105]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 2 q - 4 q + 8 q - 10 q + 12 q - 12 q + 9 q - 7 q + 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 105]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 105]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 24 26 2 q - q + 3 q + q - q + 3 q - 2 q + q - 2 q - 2 q + q - 28 30 32 34 > 2 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 105]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -2 3 z 3 z 4 z 5 z 2 z 3 z 2 z z -- + -- - --- + ---- - ---- + ---- + ---- - ---- + ---- - -- 6 4 10 8 6 4 8 6 4 6 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 105]][a, z]
Out[14]=	2 2 2 2 2 3 3 3 2 3 z z 2 z z z 2 z 8 z 6 z z 5 z 11 z -- + -- + -- - -- - --- + --- + --- - ---- - ---- - ---- - --- + ---- + ----- + 6 4 9 7 5 12 10 8 6 4 13 11 9 a a a a a a a a a a a a a 3 3 4 4 4 4 4 5 5 5 5 4 z z 7 z 4 z 8 z 8 z 3 z z 12 z 19 z 5 z > ---- - -- - ---- - ---- + ---- + ---- + ---- + --- - ----- - ----- - ---- + 7 5 12 10 8 6 4 13 11 9 7 a a a a a a a a a a a 5 6 6 6 6 7 7 7 7 8 8 z 4 z 3 z 10 z 3 z 6 z 8 z 3 z z 4 z 6 z > -- + ---- - ---- - ----- - ---- + ---- + ---- + ---- + -- + ---- + ---- + 5 12 10 8 6 11 9 7 5 10 8 a a a a a a a a a a a 8 9 9 2 z z z > ---- + -- + -- 6 9 7 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 105]], Vassiliev[3][Knot[11, NonAlternating, 105]]}
Out[15]=	{3, 5}
In[16]:=	Kh[Knot[11, NonAlternating, 105]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 11 4 2 q + q + 3 q t + q t + 5 q t + 3 q t + 5 q t + 5 q t + 7 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 5 q t + 5 q t + 7 q t + 4 q t + 5 q t + 3 q t + 19 7 19 8 21 8 23 9 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n105

K11n104 K11n106