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

K11n103 K11n105

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 X11,21,12,20 X13,1,14,22 X15,19,16,18 X17,6,18,7 X19,9,20,8 X21,13,22,12

Gauss Code:

{1, -5, 2, -1, -3, 9, -4, 10, 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 -6 -8 -12

Alexander Polynomial:

- t-4 + 3t-3 - 2t-2 - 2t-1 + 5 - 2t - 2t2 + 3t3 - t4

Conway Polynomial:

1 + z2 - 4z4 - 5z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{3, 6}

Jones Polynomial:

q - q2 + 2q3 - q4 + q5 - q7 - q9 + q10

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

{...}

A2 (sl(3)) Invariant:

q4 + q6 + q8 + 2q10 + q12 - 2q20 - q22 - 2q24 + q30 + 2q32 - q34

HOMFLY-PT Polynomial:

- a-10z2 + 2a-8 + 8a-8z2 + 6a-8z4 + a-8z6 - 6a-6 - 16a-6z2 - 16a-6z4 - 7a-6z6 - a-6z8 + 5a-4 + 10a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

- 3a-12z2 + a-12z4 + 2a-11z - 4a-11z3 + a-11z5 - 2a-10z2 + a-9z + 6a-9z3 - 6a-9z5 + a-9z7 + 2a-8 - 11a-8z2 + 22a-8z4 - 13a-8z6 + 2a-8z8 - 5a-7z + 9a-7z3 + 2a-7z5 - 5a-7z7 + a-7z9 + 6a-6 - 27a-6z2 + 39a-6z4 - 20a-6z6 + 3a-6z8 - 4a-5z - a-5z3 + 9a-5z5 - 6a-5z7 + a-5z9 + 5a-4 - 15a-4z2 + 16a-4z4 - 7a-4z6 + a-4z8

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

{1, -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=6 is the signature of 11104. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                     1 j = 19                       j = 17               1 1 1   j = 15             1 2       j = 13           1 1 1       j = 11         1 2 2         j = 9       1 1             j = 7     1 1 1             j = 5   1 2                 j = 3                       j = 1 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, 104]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 104]]
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[11, 21, 12, 20], X[13, 1, 14, 22], X[15, 19, 16, 18], > X[17, 6, 18, 7], X[19, 9, 20, 8], X[21, 13, 22, 12]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 104]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 9, -4, 10, 5, -2, -6, 11, -7, 3, -8, 4, -9, 8, -10, > 6, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 104]]
Out[5]=	DTCode[4, 10, -14, -16, 2, -20, -22, -18, -6, -8, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 104]][t]
Out[6]=	-4 3 2 2 2 3 4 5 - t + -- - -- - - - 2 t - 2 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 104]][z]
Out[7]=	2 4 6 8 1 + z - 4 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 104]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 104]], KnotSignature[Knot[11, NonAlternating, 104]]}
Out[9]=	{3, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 104]][q]
Out[10]=	2 3 4 5 7 9 10 q - q + 2 q - q + q - q - q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 104]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 104]][q]
Out[12]=	4 6 8 10 12 20 22 24 30 32 34 q + q + q + 2 q + q - 2 q - q - 2 q + q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 104]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 6 2 6 5 z 8 z 16 z 10 z 6 z 16 z 6 z z 7 z -- - -- + -- - --- + ---- - ----- + ----- + ---- - ----- + ---- + -- - ---- + 8 6 4 10 8 6 4 8 6 4 8 6 a a a a a a a a a a a a 6 8 z z > -- - -- 4 6 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 104]][a, z]
Out[14]=	2 2 2 2 2 2 6 5 2 z z 5 z 4 z 3 z 2 z 11 z 27 z 15 z -- + -- + -- + --- + -- - --- - --- - ---- - ---- - ----- - ----- - ----- - 8 6 4 11 9 7 5 12 10 8 6 4 a a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 5 4 z 6 z 9 z z z 22 z 39 z 16 z z 6 z 2 z > ---- + ---- + ---- - -- + --- + ----- + ----- + ----- + --- - ---- + ---- + 11 9 7 5 12 8 6 4 11 9 7 a a a a a a a a a a a 5 6 6 6 7 7 7 8 8 8 9 9 9 z 13 z 20 z 7 z z 5 z 6 z 2 z 3 z z z z > ---- - ----- - ----- - ---- + -- - ---- - ---- + ---- + ---- + -- + -- + -- 5 8 6 4 9 7 5 8 6 4 7 5 a a a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 104]], Vassiliev[3][Knot[11, NonAlternating, 104]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 104]][q, t]
Out[16]=	5 5 7 q q 7 9 7 2 9 2 11 2 11 3 2 q + q + -- + -- + q t + q t + q t + q t + q t + 2 q t + 2 t t 13 3 11 4 13 4 15 4 13 5 15 5 17 5 17 6 > q t + 2 q t + q t + q t + q t + 2 q t + q t + q t + 17 7 21 8 > q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n104

K11n103 K11n105