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

K11n112 K11n114

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,14,6,15 X18,7,19,8 X16,9,17,10 X2,11,3,12 X20,13,21,14 X15,22,16,1 X8,17,9,18 X12,19,13,20 X21,7,22,6

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:

- t-2 + 9t-1 - 15 + 9t - t2

Conway Polynomial:

1 + 5z2 - z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{35, -2}

Jones Polynomial:

- q-10 + 2q-9 - 4q-8 + 5q-7 - 5q-6 + 6q-5 - 5q-4 + 4q-3 - 2q-2 + q-1

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + q-28 - q-26 + q-22 - q-20 + q-18 + q-14 + q-12 + 2q-8 - q-4 + q-2

HOMFLY-PT Polynomial:

a2z2 + 2a4 + 3a4z2 - a6 - a6z2 - a6z4 + a8 + 2a8z2 - a10

Kauffman Polynomial:

a2z2 + 2a3z3 + 2a4 - 5a4z2 + 4a4z4 - 2a5z3 + a5z7 + a6 - 5a6z2 + 6a6z4 - 6a6z6 + 2a6z8 + a7z + a7z3 - 4a7z5 - a7z7 + a7z9 + a8 - 3a8z2 + 13a8z4 - 15a8z6 + 4a8z8 - 3a9z + 13a9z3 - 9a9z5 - a9z7 + a9z9 + a10 - 4a10z2 + 11a10z4 - 9a10z6 + 2a10z8 - 4a11z + 8a11z3 - 5a11z5 + a11z7

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

{5, -12}

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=-2 is the signature of 11113. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                   1 j = -3                 2 1 j = -5               2     j = -7             3 2     j = -9           3 2       j = -11         2 3         j = -13       3 3           j = -15     1 2             j = -17   1 3               j = -19   1                 j = -21 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, 113]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 113]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[18, 7, 19, 8], > X[16, 9, 17, 10], X[2, 11, 3, 12], X[20, 13, 21, 14], X[15, 22, 16, 1], > X[8, 17, 9, 18], X[12, 19, 13, 20], X[21, 7, 22, 6]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 113]]
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, 113]]
Out[5]=	DTCode[4, 10, -14, 18, 16, 2, 20, -22, 8, 12, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 113]][t]
Out[6]=	-2 9 2 -15 - t + - + 9 t - t t
In[7]:=	Conway[Knot[11, NonAlternating, 113]][z]
Out[7]=	2 4 1 + 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 113]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 113]], KnotSignature[Knot[11, NonAlternating, 113]]}
Out[9]=	{35, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 113]][q]
Out[10]=	-10 2 4 5 5 6 5 4 2 1 -q + -- - -- + -- - -- + -- - -- + -- - -- + - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 113]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 113]][q]
Out[12]=	-32 -30 -28 -26 -22 -20 -18 -14 -12 2 -4 -2 -q - q + q - q + q - q + q + q + q + -- - q + q 8 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 113]][a, z]
Out[13]=	4 6 8 10 2 2 4 2 6 2 8 2 6 4 2 a - a + a - a + a z + 3 a z - a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 113]][a, z]
Out[14]=	4 6 8 10 7 9 11 2 2 4 2 6 2 2 a + a + a + a + a z - 3 a z - 4 a z + a z - 5 a z - 5 a z - 8 2 10 2 3 3 5 3 7 3 9 3 11 3 > 3 a z - 4 a z + 2 a z - 2 a z + a z + 13 a z + 8 a z + 4 4 6 4 8 4 10 4 7 5 9 5 11 5 > 4 a z + 6 a z + 13 a z + 11 a z - 4 a z - 9 a z - 5 a z - 6 6 8 6 10 6 5 7 7 7 9 7 11 7 6 8 > 6 a z - 15 a z - 9 a z + a z - a z - a z + a z + 2 a z + 8 8 10 8 7 9 9 9 > 4 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 113]], Vassiliev[3][Knot[11, NonAlternating, 113]]}
Out[15]=	{5, -12}
In[16]:=	Kh[Knot[11, NonAlternating, 113]][q, t]
Out[16]=	-3 1 1 1 1 3 1 2 3 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q t q t q t q t q t q t q t 3 2 3 3 2 3 2 2 2 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n113

K11n112 K11n114