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

K11a149 K11a151

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 20 12 2 18 22 8 14 6

Alexander Polynomial:

- 2t-3 + 13t-2 - 29t-1 + 37 - 29t + 13t2 - 2t3

Conway Polynomial:

1 + 5z2 + z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{125, -4}

Jones Polynomial:

- q-11 + 3q-10 - 7q-9 + 12q-8 - 17q-7 + 20q-6 - 20q-5 + 18q-4 - 13q-3 + 9q-2 - 4q-1 + 1

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

{...}

A2 (sl(3)) Invariant:

- q-34 + q-30 - 3q-28 + 2q-26 + q-24 - 3q-22 + 3q-20 - 3q-18 + 2q-16 + q-14 - q-12 + 5q-10 - 3q-8 + 2q-6 + q-4 - 2q-2 + 1

HOMFLY-PT Polynomial:

a2z2 + a2z4 + 2a4 + 2a4z2 - a4z4 - a4z6 - a6 - a6z4 - a6z6 + a8 + 3a8z2 + 2a8z4 - a10 - a10z2

Kauffman Polynomial:

a2z2 - 2a2z4 + a2z6 + 4a3z3 - 9a3z5 + 4a3z7 + 2a4 - 5a4z2 + 11a4z4 - 17a4z6 + 7a4z8 + a5z5 - 10a5z7 + 6a5z9 + a6 - 9a6z2 + 27a6z4 - 31a6z6 + 9a6z8 + 2a6z10 + 2a7z - 7a7z3 + 18a7z5 - 23a7z7 + 11a7z9 + a8 - 9a8z2 + 25a8z4 - 24a8z6 + 8a8z8 + 2a8z10 + 2a9z3 - 4a9z7 + 5a9z9 + a10 - 4a10z2 + 6a10z4 - 8a10z6 + 6a10z8 - a11z + 3a11z3 - 7a11z5 + 5a11z7 + 2a12z2 - 5a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

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=-4 is the signature of 11150. 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 r = 1 r = 2 j = 1                       1 j = -1                     3   j = -3                   6 1   j = -5                 8 4     j = -7               10 5       j = -9             10 8         j = -11           10 10           j = -13         7 10             j = -15       5 10               j = -17     2 7                 j = -19   1 5                   j = -21   2                     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, Alternating, 150]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 150]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[20, 7, 21, 8], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[22, 15, 1, 16], > X[8, 17, 9, 18], X[14, 19, 15, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 150]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -5, 7, -10, 8, -3, 9, -7, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 150]]
Out[5]=	DTCode[4, 10, 16, 20, 12, 2, 18, 22, 8, 14, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 150]][t]
Out[6]=	2 13 29 2 3 37 - -- + -- - -- - 29 t + 13 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 150]][z]
Out[7]=	2 4 6 1 + 5 z + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 150]}
In[9]:=	{KnotDet[Knot[11, Alternating, 150]], KnotSignature[Knot[11, Alternating, 150]]}
Out[9]=	{125, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 150]][q]
Out[10]=	-11 3 7 12 17 20 20 18 13 9 4 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q 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, Alternating, 150]}
In[12]:=	A2Invariant[Knot[11, Alternating, 150]][q]
Out[12]=	-34 -30 3 2 -24 3 3 3 2 -14 -12 1 - q + q - --- + --- + q - --- + --- - --- + --- + q - q + 28 26 22 20 18 16 q q q q q q 5 3 2 -4 2 > --- - -- + -- + q - -- 10 8 6 2 q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 150]][a, z]
Out[13]=	4 6 8 10 2 2 4 2 8 2 10 2 2 4 4 4 2 a - a + a - a + a z + 2 a z + 3 a z - a z + a z - a z - 6 4 8 4 4 6 6 6 > a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 150]][a, z]
Out[14]=	4 6 8 10 7 11 13 2 2 4 2 6 2 2 a + a + a + a + 2 a z - a z + a z + a z - 5 a z - 9 a z - 8 2 10 2 12 2 3 3 7 3 9 3 11 3 > 9 a z - 4 a z + 2 a z + 4 a z - 7 a z + 2 a z + 3 a z - 13 3 2 4 4 4 6 4 8 4 10 4 12 4 > 2 a z - 2 a z + 11 a z + 27 a z + 25 a z + 6 a z - 5 a z - 3 5 5 5 7 5 11 5 13 5 2 6 4 6 > 9 a z + a z + 18 a z - 7 a z + a z + a z - 17 a z - 6 6 8 6 10 6 12 6 3 7 5 7 7 7 > 31 a z - 24 a z - 8 a z + 3 a z + 4 a z - 10 a z - 23 a z - 9 7 11 7 4 8 6 8 8 8 10 8 5 9 > 4 a z + 5 a z + 7 a z + 9 a z + 8 a z + 6 a z + 6 a z + 7 9 9 9 6 10 8 10 > 11 a z + 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 150]], Vassiliev[3][Knot[11, Alternating, 150]]}
Out[15]=	{5, -12}
In[16]:=	Kh[Knot[11, Alternating, 150]][q, t]
Out[16]=	4 6 1 2 1 5 2 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 10 7 10 10 10 10 8 10 5 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 8 t 3 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a150

K11a149 K11a151