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

K11a111 K11a113

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 18 2 20 6 22 12 8

Alexander Polynomial:

t-4 - 6t-3 + 15t-2 - 25t-1 + 31 - 25t + 15t2 - 6t3 + t4

Conway Polynomial:

1 - 3z2 - z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{125, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 12q-3 + 17q-2 - 20q-1 + 20 - 18q + 14q2 - 8q3 + 4q4 - q5

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

{K11a5, ...}

A2 (sl(3)) Invariant:

q-18 + 2q-12 - 3q-10 + 3q-8 - q-6 - 2q-4 + 2q-2 - 5 + 4q2 - 2q4 + 2q6 + 3q8 - 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

a-2 - 2a-2z2 - 3a-2z4 - a-2z6 + 1 + 6z2 + 9z4 + 5z6 + z8 - 3a2 - 10a2z2 - 8a2z4 - 2a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 - a-3z + 3a-3z3 - 10a-3z5 + 7a-3z7 - a-2 + 3a-2z2 - 10a-2z6 + 8a-2z8 - 3a-1z + 8a-1z3 - 6a-1z5 - 4a-1z7 + 6a-1z9 + 1 - 10z2 + 30z4 - 30z6 + 10z8 + 2z10 - az - 3az3 + 19az5 - 24az7 + 11az9 + 3a2 - 21a2z2 + 39a2z4 - 30a2z6 + 7a2z8 + 2a2z10 - 2a3z3 + 6a3z5 - 10a3z7 + 5a3z9 + 2a4 - 8a4z2 + 12a4z4 - 13a4z6 + 5a4z8 - a5z + 5a5z3 - 8a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{-3, 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=0 is the signature of 11112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   5 1   j = 5                 9 3     j = 3               9 5       j = 1             11 9         j = -1           10 10           j = -3         7 10             j = -5       5 10               j = -7     2 7                 j = -9   1 5                   j = -11   2                     j = -13 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, 112]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 112]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[16, 8, 17, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 13, 21, 14], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 112]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -11, 5, -2, 6, -10, 7, -3, 8, -4, 9, -5, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 112]]
Out[5]=	DTCode[4, 10, 14, 16, 18, 2, 20, 6, 22, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 112]][t]
Out[6]=	-4 6 15 25 2 3 4 31 + t - -- + -- - -- - 25 t + 15 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 112]][z]
Out[7]=	2 4 6 8 1 - 3 z - z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 112]}
In[9]:=	{KnotDet[Knot[11, Alternating, 112]], KnotSignature[Knot[11, Alternating, 112]]}
Out[9]=	{125, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 112]][q]
Out[10]=	-6 3 7 12 17 20 2 3 4 5 20 + q - -- + -- - -- + -- - -- - 18 q + 14 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 5], Knot[11, Alternating, 112]}
In[12]:=	A2Invariant[Knot[11, Alternating, 112]][q]
Out[12]=	-18 2 3 3 -6 2 2 2 4 6 8 -5 + q + --- - --- + -- - q - -- + -- + 4 q - 2 q + 2 q + 3 q - 12 10 8 4 2 q q q q q 10 12 14 > 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 112]][a, z]
Out[13]=	2 4 -2 2 4 2 2 z 2 2 4 2 4 3 z 1 + a - 3 a + 2 a + 6 z - ---- - 10 a z + 3 a z + 9 z - ---- - 2 2 a a 6 2 4 4 4 6 z 2 6 8 > 8 a z + a z + 5 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 112]][a, z]
Out[14]=	2 2 -2 2 4 z 3 z 5 2 2 z 3 z 1 - a + 3 a + 2 a - -- - --- - a z - a z - 10 z + ---- + ---- - 3 a 4 2 a a a 3 3 3 2 2 4 2 6 2 z 3 z 8 z 3 3 3 > 21 a z - 8 a z + 2 a z - -- + ---- + ---- - 3 a z - 2 a z + 5 3 a a a 4 5 5 5 3 4 6 z 2 4 4 4 6 4 z 10 z > 5 a z + 30 z - ---- + 39 a z + 12 a z - 3 a z + -- - ----- - 4 5 3 a a a 5 6 6 6 z 5 3 5 5 5 6 4 z 10 z 2 6 > ---- + 19 a z + 6 a z - 8 a z - 30 z + ---- - ----- - 30 a z - a 4 2 a a 7 7 4 6 6 6 7 z 4 z 7 3 7 5 7 8 > 13 a z + a z + ---- - ---- - 24 a z - 10 a z + 3 a z + 10 z + 3 a a 8 9 8 z 2 8 4 8 6 z 9 3 9 10 2 10 > ---- + 7 a z + 5 a z + ---- + 11 a z + 5 a z + 2 z + 2 a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 112]], Vassiliev[3][Knot[11, Alternating, 112]]}
Out[15]=	{-3, 2}
In[16]:=	Kh[Knot[11, Alternating, 112]][q, t]
Out[16]=	10 1 2 1 5 2 7 5 10 -- + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 7 10 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 9 q t + 5 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a112

K11a111 K11a113