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

K11n108 K11n110

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,14,6,15 X16,7,17,8 X18,9,19,10 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, -4, 9, -5, 10, -7, -11, 8}

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 13t-1 + 15 - 13t + 7t2 - t3

Conway Polynomial:

1 + 6z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n137, ...}

Determinant and Signature:

{57, -4}

Jones Polynomial:

- q-11 + 2q-10 - 5q-9 + 8q-8 - 9q-7 + 10q-6 - 9q-5 + 7q-4 - 4q-3 + 2q-2

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

{...}

A2 (sl(3)) Invariant:

- q-34 - q-32 - 2q-28 + 2q-26 + q-24 + 2q-20 - 2q-18 + 2q-16 - q-14 + 2q-10 - q-8 + 2q-6

HOMFLY-PT Polynomial:

2a4 + 5a4z2 + 2a4z4 - 2a6 - 3a6z2 - 3a6z4 - a6z6 + 3a8 + 5a8z2 + 2a8z4 - 2a10 - a10z2

Kauffman Polynomial:

2a4 - 6a4z2 + 3a4z4 - 3a5z3 + a5z5 + a5z7 + 2a6 - 8a6z2 + 8a6z4 - 4a6z6 + 2a6z8 + 2a7z3 - 2a7z5 + a7z7 + a7z9 + 3a8 - 13a8z2 + 20a8z4 - 13a8z6 + 5a8z8 - 4a9z + 12a9z3 - 10a9z5 + 3a9z7 + a9z9 + 2a10 - 10a10z2 + 11a10z4 - 7a10z6 + 3a10z8 - 2a11z + 4a11z3 - 6a11z5 + 3a11z7 + a12z2 - 4a12z4 + 2a12z6 + 2a13z - 3a13z3 + a13z5

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

{6, -16}

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 11109. 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 = -3                   2 j = -5                 3 1 j = -7               4 1   j = -9             5 3     j = -11           5 4       j = -13         4 5         j = -15       4 5           j = -17     1 4             j = -19   1 4               j = -21   1                 j = -23 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, 109]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 109]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[16, 7, 17, 8], > X[18, 9, 19, 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, 109]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 11, 4, -9, 5, -2, 6, -10, 7, 3, -8, -4, 9, -5, 10, > -7, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 109]]
Out[5]=	DTCode[4, 10, -14, 16, 18, 2, 20, -22, 8, 12, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 109]][t]
Out[6]=	-3 7 13 2 3 15 - t + -- - -- - 13 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 109]][z]
Out[7]=	2 4 6 1 + 6 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 109], Knot[11, NonAlternating, 137]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 109]], KnotSignature[Knot[11, NonAlternating, 109]]}
Out[9]=	{57, -4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 109]][q]
Out[10]=	-11 2 5 8 9 10 9 7 4 2 -q + --- - -- + -- - -- + -- - -- + -- - -- + -- 10 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, 109]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 109]][q]
Out[12]=	-34 -32 2 2 -24 2 2 2 -14 2 -8 2 -q - q - --- + --- + q + --- - --- + --- - q + --- - q + -- 28 26 20 18 16 10 6 q q q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 109]][a, z]
Out[13]=	4 6 8 10 4 2 6 2 8 2 10 2 4 4 2 a - 2 a + 3 a - 2 a + 5 a z - 3 a z + 5 a z - a z + 2 a z - 6 4 8 4 6 6 > 3 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 109]][a, z]
Out[14]=	4 6 8 10 9 11 13 4 2 6 2 2 a + 2 a + 3 a + 2 a - 4 a z - 2 a z + 2 a z - 6 a z - 8 a z - 8 2 10 2 12 2 5 3 7 3 9 3 11 3 > 13 a z - 10 a z + a z - 3 a z + 2 a z + 12 a z + 4 a z - 13 3 4 4 6 4 8 4 10 4 12 4 5 5 > 3 a z + 3 a z + 8 a z + 20 a z + 11 a z - 4 a z + a z - 7 5 9 5 11 5 13 5 6 6 8 6 10 6 > 2 a z - 10 a z - 6 a z + a z - 4 a z - 13 a z - 7 a z + 12 6 5 7 7 7 9 7 11 7 6 8 8 8 > 2 a z + a z + a z + 3 a z + 3 a z + 2 a z + 5 a z + 10 8 7 9 9 9 > 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 109]], Vassiliev[3][Knot[11, NonAlternating, 109]]}
Out[15]=	{6, -16}
In[16]:=	Kh[Knot[11, NonAlternating, 109]][q, t]
Out[16]=	-5 2 1 1 1 4 1 4 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q t q t q t q t q t q t q t 5 4 5 5 4 5 3 4 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 3 > ---- 5 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n109

K11n108 K11n110