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

K11a143 K11a145

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 18 12 2 22 20 6 8 14

Alexander Polynomial:

- 2t-3 + 10t-2 - 16t-1 + 17 - 16t + 10t2 - 2t3

Conway Polynomial:

1 + 6z2 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{73, -4}

Jones Polynomial:

- q-11 + 2q-10 - 5q-9 + 8q-8 - 10q-7 + 12q-6 - 11q-5 + 10q-4 - 7q-3 + 4q-2 - 2q-1 + 1

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

{...}

A2 (sl(3)) Invariant:

- q-34 - q-32 - 2q-28 + q-26 + q-24 + 3q-20 - q-18 + 2q-16 - q-12 + 2q-10 - 2q-8 + q-6 + 1

HOMFLY-PT Polynomial:

a2 + 3a2z2 + a2z4 - a4 - 2a4z2 - 3a4z4 - a4z6 + a6 + a6z2 - 2a6z4 - a6z6 + 2a8 + 5a8z2 + 2a8z4 - 2a10 - a10z2

Kauffman Polynomial:

- a2 + 4a2z2 - 4a2z4 + a2z6 - a3z + 6a3z3 - 7a3z5 + 2a3z7 - a4 + 4a4z2 - 2a4z4 - 4a4z6 + 2a4z8 - a5z - 3a5z3 + 5a5z5 - 5a5z7 + 2a5z9 - a6 + 4a6z2 - 5a6z4 + 2a6z6 - a6z8 + a6z10 + a7z - 11a7z3 + 19a7z5 - 12a7z7 + 4a7z9 + 2a8 - 4a8z2 + 5a8z4 - a8z6 + a8z10 - 3a9z + 5a9z3 - 2a9z7 + 2a9z9 + 2a10 - 7a10z2 + 8a10z4 - 6a10z6 + 3a10z8 - 2a11z + 4a11z3 - 6a11z5 + 3a11z7 + a12z2 - 4a12z4 + 2a12z6 + 2a13z - 3a13z3 + a13z5

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

{6, -17}

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 11144. 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                     1   j = -3                   3 1   j = -5                 5 2     j = -7               5 2       j = -9             6 5         j = -11           6 5           j = -13         4 6             j = -15       4 6               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, Alternating, 144]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 144]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[18, 7, 19, 8], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[22, 13, 1, 14], X[20, 15, 21, 16], > X[6, 17, 7, 18], X[8, 19, 9, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 144]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -5, 7, -11, 8, -3, 9, -4, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 144]]
Out[5]=	DTCode[4, 10, 16, 18, 12, 2, 22, 20, 6, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 144]][t]
Out[6]=	2 10 16 2 3 17 - -- + -- - -- - 16 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 144]][z]
Out[7]=	2 4 6 1 + 6 z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 144]}
In[9]:=	{KnotDet[Knot[11, Alternating, 144]], KnotSignature[Knot[11, Alternating, 144]]}
Out[9]=	{73, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 144]][q]
Out[10]=	-11 2 5 8 10 12 11 10 7 4 2 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, 144]}
In[12]:=	A2Invariant[Knot[11, Alternating, 144]][q]
Out[12]=	-34 -32 2 -26 -24 3 -18 2 -12 2 2 -6 1 - q - q - --- + q + q + --- - q + --- - q + --- - -- + q 28 20 16 10 8 q q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 144]][a, z]
Out[13]=	2 4 6 8 10 2 2 4 2 6 2 8 2 10 2 a - a + a + 2 a - 2 a + 3 a z - 2 a z + a z + 5 a z - a z + 2 4 4 4 6 4 8 4 4 6 6 6 > a z - 3 a z - 2 a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 144]][a, z]
Out[14]=	2 4 6 8 10 3 5 7 9 11 -a - a - a + 2 a + 2 a - a z - a z + a z - 3 a z - 2 a z + 13 2 2 4 2 6 2 8 2 10 2 12 2 > 2 a z + 4 a z + 4 a z + 4 a z - 4 a z - 7 a z + a z + 3 3 5 3 7 3 9 3 11 3 13 3 2 4 > 6 a z - 3 a z - 11 a z + 5 a z + 4 a z - 3 a z - 4 a z - 4 4 6 4 8 4 10 4 12 4 3 5 5 5 > 2 a z - 5 a z + 5 a z + 8 a z - 4 a z - 7 a z + 5 a z + 7 5 11 5 13 5 2 6 4 6 6 6 8 6 > 19 a z - 6 a z + a z + a z - 4 a z + 2 a z - a z - 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 6 a z + 2 a z + 2 a z - 5 a z - 12 a z - 2 a z + 3 a z + 4 8 6 8 10 8 5 9 7 9 9 9 6 10 8 10 > 2 a z - a z + 3 a z + 2 a z + 4 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 144]], Vassiliev[3][Knot[11, Alternating, 144]]}
Out[15]=	{6, -17}
In[16]:=	Kh[Knot[11, Alternating, 144]][q, t]
Out[16]=	2 3 1 1 1 4 1 4 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 6 4 6 6 5 6 5 5 2 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 5 t t 2 > ---- + -- + - + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a144

K11a143 K11a145