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

K11a305 K11a307

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X12,3,13,4 X14,5,15,6 X16,8,17,7 X18,9,19,10 X20,11,21,12 X4,13,5,14 X2,15,3,16 X22,18,1,17 X10,19,11,20 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 14 16 18 20 4 2 22 10 8

Alexander Polynomial:

t-4 - 5t-3 + 13t-2 - 21t-1 + 25 - 21t + 13t2 - 5t3 + t4

Conway Polynomial:

1 + 2z2 + 3z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a175, ...}

Determinant and Signature:

{105, -4}

Jones Polynomial:

q-10 - 4q-9 + 8q-8 - 12q-7 + 15q-6 - 17q-5 + 16q-4 - 13q-3 + 10q-2 - 5q-1 + 3 - q

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

{...}

A2 (sl(3)) Invariant:

q-30 - q-28 + q-24 - 3q-22 + 2q-20 - 2q-18 + q-14 - 3q-12 + 4q-10 - q-8 + 3q-6 + 2q-4 - q-2 + 1 - q2

HOMFLY-PT Polynomial:

- 4a2z2 - 4a2z4 - a2z6 + 4a4 + 14a4z2 + 14a4z4 + 6a4z6 + a4z8 - 4a6 - 10a6z2 - 8a6z4 - 2a6z6 + a8 + 2a8z2 + a8z4

Kauffman Polynomial:

4az3 - 4az5 + az7 - 8a2z2 + 18a2z4 - 13a2z6 + 3a2z8 + a3z + 6a3z5 - 9a3z7 + 3a3z9 + 4a4 - 21a4z2 + 39a4z4 - 30a4z6 + 6a4z8 + a4z10 - a5z + 3a5z3 + 4a5z5 - 15a5z7 + 7a5z9 + 4a6 - 14a6z2 + 27a6z4 - 32a6z6 + 11a6z8 + a6z10 - 3a7z + 15a7z3 - 23a7z5 + 5a7z7 + 4a7z9 + a8 + a8z2 - 3a8z4 - 7a8z6 + 8a8z8 - a9z + 6a9z3 - 13a9z5 + 10a9z7 + 2a10z2 - 8a10z4 + 8a10z6 - 2a11z3 + 4a11z5 + a12z4

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

{2, -2}

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 11306. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 3                       1 j = 1                     2   j = -1                   3 1   j = -3                 7 2     j = -5               7 4       j = -7             9 6         j = -9           8 7           j = -11         7 9             j = -13       5 8               j = -15     3 7                 j = -17   1 5                   j = -19   3                     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, Alternating, 306]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 306]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 3, 13, 4], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[18, 9, 19, 10], X[20, 11, 21, 12], X[4, 13, 5, 14], X[2, 15, 3, 16], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 306]]
Out[4]=	GaussCode[1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 306]]
Out[5]=	DTCode[6, 12, 14, 16, 18, 20, 4, 2, 22, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 306]][t]
Out[6]=	-4 5 13 21 2 3 4 25 + t - -- + -- - -- - 21 t + 13 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 306]][z]
Out[7]=	2 4 6 8 1 + 2 z + 3 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 175], Knot[11, Alternating, 306]}
In[9]:=	{KnotDet[Knot[11, Alternating, 306]], KnotSignature[Knot[11, Alternating, 306]]}
Out[9]=	{105, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 306]][q]
Out[10]=	-10 4 8 12 15 17 16 13 10 5 3 + q - -- + -- - -- + -- - -- + -- - -- + -- - - - 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, Alternating, 306]}
In[12]:=	A2Invariant[Knot[11, Alternating, 306]][q]
Out[12]=	-30 -28 -24 3 2 2 -14 3 4 -8 3 2 1 + q - q + q - --- + --- - --- + q - --- + --- - q + -- + -- - 22 20 18 12 10 6 4 q q q q q q q -2 2 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 306]][a, z]
Out[13]=	4 6 8 2 2 4 2 6 2 8 2 2 4 4 a - 4 a + a - 4 a z + 14 a z - 10 a z + 2 a z - 4 a z + 4 4 6 4 8 4 2 6 4 6 6 6 4 8 > 14 a z - 8 a z + a z - a z + 6 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 306]][a, z]
Out[14]=	4 6 8 3 5 7 9 2 2 4 2 4 a + 4 a + a + a z - a z - 3 a z - a z - 8 a z - 21 a z - 6 2 8 2 10 2 3 5 3 7 3 9 3 > 14 a z + a z + 2 a z + 4 a z + 3 a z + 15 a z + 6 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 2 a z + 18 a z + 39 a z + 27 a z - 3 a z - 8 a z + a z - 5 3 5 5 5 7 5 9 5 11 5 2 6 > 4 a z + 6 a z + 4 a z - 23 a z - 13 a z + 4 a z - 13 a z - 4 6 6 6 8 6 10 6 7 3 7 5 7 > 30 a z - 32 a z - 7 a z + 8 a z + a z - 9 a z - 15 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 5 a z + 10 a z + 3 a z + 6 a z + 11 a z + 8 a z + 3 a z + 5 9 7 9 4 10 6 10 > 7 a z + 4 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 306]], Vassiliev[3][Knot[11, Alternating, 306]]}
Out[15]=	{2, -2}
In[16]:=	Kh[Knot[11, Alternating, 306]][q, t]
Out[16]=	4 7 1 3 1 5 3 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 8 7 9 8 7 9 6 7 2 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q t q t q t q t q t q t q t q t q 2 3 t t 2 3 3 > --- + -- + 2 q t + q t q q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a306

K11a305 K11a307