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

K11a104 K11a106

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X16,8,17,7 X2,10,3,9 X20,11,21,12 X8,14,9,13 X6,16,7,15 X22,18,1,17 X12,19,13,20 X18,22,19,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 2 20 8 6 22 12 18

Alexander Polynomial:

- 3t-3 + 13t-2 - 24t-1 + 29 - 24t + 13t2 - 3t3

Conway Polynomial:

1 + z2 - 5z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{109, 4}

Jones Polynomial:

1 - 2q + 6q2 - 10q3 + 14q4 - 17q5 + 18q6 - 16q7 + 12q8 - 8q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 + q4 + 3q6 - 2q8 + 3q10 - 2q12 - 2q14 + 2q16 - 3q18 + 3q20 - 2q22 + 2q26 - 2q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 3a-8 + 7a-8z2 + 3a-8z4 - 3a-6 - 8a-6z2 - 7a-6z4 - 2a-6z6 - 2a-4z4 - a-4z6 + 2a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + a-12z2 - 6a-12z4 + 4a-12z6 - a-11z + 5a-11z3 - 12a-11z5 + 7a-11z7 + a-10 - 2a-10z2 + 4a-10z4 - 10a-10z6 + 7a-10z8 - a-9z + 11a-9z3 - 14a-9z5 + 2a-9z7 + 4a-9z9 + 3a-8 - 15a-8z2 + 33a-8z4 - 30a-8z6 + 11a-8z8 + a-8z10 + a-7z - 4a-7z3 + 9a-7z5 - 12a-7z7 + 7a-7z9 + 3a-6 - 17a-6z2 + 29a-6z4 - 23a-6z6 + 7a-6z8 + a-6z10 + 2a-5z - 7a-5z3 + 5a-5z5 - 5a-5z7 + 3a-5z9 + 2a-4z4 - 6a-4z6 + 3a-4z8 + a-3z + 2a-3z3 - 5a-3z5 + 2a-3z7 - 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6

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

{1, 2}

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 = -2 r = -1 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                   5 1   j = 17                 7 3     j = 15               9 5       j = 13             9 7         j = 11           8 9           j = 9         6 9             j = 7       4 8               j = 5     2 6                 j = 3   1 5                   j = 1   1                     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, Alternating, 105]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 105]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[8, 14, 9, 13], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 105]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -7, 5, -2, 6, -10, 7, -3, 8, -4, 9, -11, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 105]]
Out[5]=	DTCode[4, 10, 14, 16, 2, 20, 8, 6, 22, 12, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 105]][t]
Out[6]=	3 13 24 2 3 29 - -- + -- - -- - 24 t + 13 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 105]][z]
Out[7]=	2 4 6 1 + z - 5 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 105]}
In[9]:=	{KnotDet[Knot[11, Alternating, 105]], KnotSignature[Knot[11, Alternating, 105]]}
Out[9]=	{109, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 105]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 6 q - 10 q + 14 q - 17 q + 18 q - 16 q + 12 q - 8 q + 4 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 105]}
In[12]:=	A2Invariant[Knot[11, Alternating, 105]][q]
Out[12]=	4 6 8 10 12 14 16 18 20 22 1 + q + 3 q - 2 q + 3 q - 2 q - 2 q + 2 q - 3 q + 3 q - 2 q + 26 28 30 34 > 2 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 105]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -10 3 3 2 z 7 z 8 z 3 z 3 z 7 z 2 z z -a + -- - -- + -- - --- + ---- - ---- + ---- + ---- - ---- - ---- + -- - 8 6 2 10 8 6 2 8 6 4 2 a a a a a a a a a a a 6 6 2 z z > ---- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 105]][a, z]
Out[14]=	2 2 2 2 -10 3 3 2 z z z 2 z z z 2 z 15 z 17 z a + -- + -- - -- - --- - -- + -- + --- + -- + --- - ---- - ----- - ----- + 8 6 2 11 9 7 5 3 12 10 8 6 a a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 5 z z 5 z 11 z 4 z 7 z 2 z 6 z 4 z 33 z > ---- - --- + ---- + ----- - ---- - ---- + ---- - ---- + ---- + ----- + 2 13 11 9 7 5 3 12 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 29 z 2 z 4 z z 12 z 14 z 9 z 5 z 5 z 4 z > ----- + ---- - ---- + --- - ----- - ----- + ---- + ---- - ---- + ---- - 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 10 z 30 z 23 z 6 z z 7 z 2 z 12 z 5 z 2 z > ----- - ----- - ----- - ---- + -- + ---- + ---- - ----- - ---- + ---- + 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 7 z 11 z 7 z 3 z 4 z 7 z 3 z z z > ---- + ----- + ---- + ---- + ---- + ---- + ---- + --- + --- 10 8 6 4 9 7 5 8 6 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 105]], Vassiliev[3][Knot[11, Alternating, 105]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, Alternating, 105]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 5 q + 2 q + ---- + - + -- + 6 q t + 4 q t + 8 q t + 6 q t + 9 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 8 q t + 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 7 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a105

K11a104 K11a106