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

K11n113 K11n115

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,6,15,5 X18,7,19,8 X16,9,17,10 X2,11,3,12 X13,21,14,20 X22,16,1,15 X8,17,9,18 X19,13,20,12 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 16 2 -20 22 8 -12 6

Alexander Polynomial:

3t-2 - 13t-1 + 21 - 13t + 3t2

Conway Polynomial:

1 - z2 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{K11a195, ...}

Determinant and Signature:

{53, 0}

Jones Polynomial:

q-4 - 3q-3 + 6q-2 - 8q-1 + 9 - 9q + 8q2 - 5q3 + 3q4 - q5

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

{930, ...}

A2 (sl(3)) Invariant:

q-14 + q-12 - 2q-10 + q-8 - 2q-4 + 2q-2 - 1 + 2q2 + 2q8 - 2q10 + q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z4 + 2 + 3z2 + 2z4 - 2a2 - 3a2z2 + a4

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 3a-4z2 - 7a-4z4 + 3a-4z6 - a-3z + 5a-3z3 - 9a-3z5 + 4a-3z7 + 2a-2z4 - 5a-2z6 + 3a-2z8 - 2a-1z + 6a-1z3 - 6a-1z5 + 2a-1z7 + a-1z9 + 2 - 10z2 + 16z4 - 10z6 + 4z8 - 5az3 + 7az5 - 2az7 + az9 + 2a2 - 9a2z2 + 8a2z4 - 2a2z6 + a2z8 + a3z - 4a3z3 + 3a3z5 + a4 - 2a4z2 + a4z4

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 11114. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj 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                 2   j = 7               3 1   j = 5             5 2     j = 3           4 3       j = 1         5 5         j = -1       4 5           j = -3     2 4             j = -5   1 4               j = -7   2                 j = -9 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, 114]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 114]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[18, 7, 19, 8], > X[16, 9, 17, 10], X[2, 11, 3, 12], X[13, 21, 14, 20], X[22, 16, 1, 15], > X[8, 17, 9, 18], X[19, 13, 20, 12], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 114]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, 10, -7, -3, 8, -5, 9, -4, -10, > 7, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 114]]
Out[5]=	DTCode[4, 10, 14, 18, 16, 2, -20, 22, 8, -12, 6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 114]][t]
Out[6]=	3 13 2 21 + -- - -- - 13 t + 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 114]][z]
Out[7]=	2 4 1 - z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 195], Knot[11, NonAlternating, 114]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 114]], KnotSignature[Knot[11, NonAlternating, 114]]}
Out[9]=	{53, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 114]][q]
Out[10]=	-4 3 6 8 2 3 4 5 9 + q - -- + -- - - - 9 q + 8 q - 5 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 30], Knot[11, NonAlternating, 114]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 114]][q]
Out[12]=	-14 -12 2 -8 2 2 2 8 10 12 14 16 -1 + q + q - --- + q - -- + -- + 2 q + 2 q - 2 q + q + q - q 10 4 2 q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 114]][a, z]
Out[13]=	2 4 2 4 2 z 2 2 4 z 2 - 2 a + a + 3 z - -- - 3 a z + 2 z + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 114]][a, z]
Out[14]=	2 3 2 4 z 2 z 3 2 3 z 2 2 4 2 2 z 2 + 2 a + a - -- - --- + a z - 10 z + ---- - 9 a z - 2 a z - ---- + 3 a 4 5 a a a 3 3 4 4 5 z 6 z 3 3 3 4 7 z 2 z 2 4 4 4 > ---- + ---- - 5 a z - 4 a z + 16 z - ---- + ---- + 8 a z + a z + 3 a 4 2 a a a 5 5 5 6 6 z 9 z 6 z 5 3 5 6 3 z 5 z 2 6 > -- - ---- - ---- + 7 a z + 3 a z - 10 z + ---- - ---- - 2 a z + 5 3 a 4 2 a a a a 7 7 8 9 4 z 2 z 7 8 3 z 2 8 z 9 > ---- + ---- - 2 a z + 4 z + ---- + a z + -- + a z 3 a 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 114]], Vassiliev[3][Knot[11, NonAlternating, 114]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 114]][q, t]
Out[16]=	5 1 2 1 4 2 4 4 3 - + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 5 q t + 4 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 3 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n114

K11n113 K11n115