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

K11a105 K11a107

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 2 20 18 6 8 22 12

Alexander Polynomial:

t-4 - 5t-3 + 12t-2 - 18t-1 + 21 - 18t + 12t2 - 5t3 + t4

Conway Polynomial:

1 + z2 + 2z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a194, K11a346, ...}

Determinant and Signature:

{93, 0}

Jones Polynomial:

- q-5 + 3q-4 - 6q-3 + 10q-2 - 13q-1 + 15 - 14q + 13q2 - 9q3 + 5q4 - 3q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + q-12 - 2q-10 + q-8 + q-6 - q-4 + 4q-2 - 2 + 3q2 + 2q8 - 3q10 - q14 + q18

HOMFLY-PT Polynomial:

a-4 + 3a-4z2 + a-4z4 - 4a-2 - 12a-2z2 - 9a-2z4 - 2a-2z6 + 6 + 15z2 + 14z4 + 6z6 + z8 - 2a2 - 5a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 3a-6z4 + a-6z6 - 2a-5z + 8a-5z3 - 10a-5z5 + 3a-5z7 + a-4 - 5a-4z2 + 10a-4z4 - 12a-4z6 + 4a-4z8 - 6a-3z + 17a-3z3 - 9a-3z5 - 4a-3z7 + 3a-3z9 + 4a-2 - 22a-2z2 + 39a-2z4 - 27a-2z6 + 6a-2z8 + a-2z10 - 6a-1z + 9a-1z3 + 7a-1z5 - 13a-1z7 + 6a-1z9 + 6 - 23z2 + 33z4 - 21z6 + 6z8 + z10 - 2az + az3 - 2az7 + 3az9 + 2a2 - 4a2z2 + a2z4 - 4a2z6 + 4a2z8 + a3z - a3z3 - 5a3z5 + 4a3z7 + 3a4z2 - 6a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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=0 is the signature of 11106. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     2   j = 9                   3 1   j = 7                 6 2     j = 5               7 3       j = 3             7 6         j = 1           8 7           j = -1         6 8             j = -3       4 7               j = -5     2 6                 j = -7   1 4                   j = -9   2                     j = -11 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, 106]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 106]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[16, 7, 17, 8], > X[2, 10, 3, 9], X[20, 12, 21, 11], X[18, 13, 19, 14], X[6, 15, 7, 16], > X[8, 17, 9, 18], X[22, 20, 1, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 106]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -7, 10, > -6, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 106]]
Out[5]=	DTCode[4, 10, 14, 16, 2, 20, 18, 6, 8, 22, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 106]][t]
Out[6]=	-4 5 12 18 2 3 4 21 + t - -- + -- - -- - 18 t + 12 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 106]][z]
Out[7]=	2 4 6 8 1 + z + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 106], Knot[11, Alternating, 194], > Knot[11, Alternating, 346]}
In[9]:=	{KnotDet[Knot[11, Alternating, 106]], KnotSignature[Knot[11, Alternating, 106]]}
Out[9]=	{93, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 106]][q]
Out[10]=	-5 3 6 10 13 2 3 4 5 6 15 - q + -- - -- + -- - -- - 14 q + 13 q - 9 q + 5 q - 3 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, Alternating, 106]}
In[12]:=	A2Invariant[Knot[11, Alternating, 106]][q]
Out[12]=	-14 -12 2 -8 -6 -4 4 2 8 10 14 18 -2 - q + q - --- + q + q - q + -- + 3 q + 2 q - 3 q - q + q 10 2 q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 106]][a, z]
Out[13]=	2 2 4 4 -4 4 2 2 3 z 12 z 2 2 4 z 9 z 6 + a - -- - 2 a + 15 z + ---- - ----- - 5 a z + 14 z + -- - ---- - 2 4 2 4 2 a a a a a 6 2 4 6 2 z 2 6 8 > 4 a z + 6 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 106]][a, z]
Out[14]=	2 -4 4 2 2 z 6 z 6 z 3 5 2 z 6 + a + -- + 2 a - --- - --- - --- - 2 a z + a z + a z - 23 z + -- - 2 5 3 a 6 a a a a 2 2 3 3 3 5 z 22 z 2 2 4 2 8 z 17 z 9 z 3 3 3 > ---- - ----- - 4 a z + 3 a z + ---- + ----- + ---- + a z - a z - 4 2 5 3 a a a a a 4 4 4 5 5 5 3 4 3 z 10 z 39 z 2 4 4 4 10 z 9 z > 2 a z + 33 z - ---- + ----- + ----- + a z - 6 a z - ----- - ---- + 6 4 2 5 3 a a a a a 5 6 6 6 7 z 3 5 5 5 6 z 12 z 27 z 2 6 4 6 > ---- - 5 a z + a z - 21 z + -- - ----- - ----- - 4 a z + 3 a z + a 6 4 2 a a a 7 7 7 8 8 3 z 4 z 13 z 7 3 7 8 4 z 6 z 2 8 > ---- - ---- - ----- - 2 a z + 4 a z + 6 z + ---- + ---- + 4 a z + 5 3 a 4 2 a a a a 9 9 10 3 z 6 z 9 10 z > ---- + ---- + 3 a z + z + --- 3 a 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 106]], Vassiliev[3][Knot[11, Alternating, 106]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, Alternating, 106]][q, t]
Out[16]=	8 1 2 1 4 2 6 4 7 6 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 > 7 q t + 7 q t + 6 q t + 7 q t + 3 q t + 6 q t + 2 q t + 9 4 9 5 11 5 13 6 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a106

K11a105 K11a107