10 132

10_131

10_133

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10 132 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X8493 X5,12,6,13 X15,18,16,19 X9,16,10,17 X17,10,18,11 X13,20,14,1 X19,14,20,15 X11,6,12,7 X2837
Gauss code	1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7
Dowker-Thistlethwaite code	4 8 -12 2 -16 -6 -20 -18 -10 -14
Conway Notation	[23,3,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-1] Hyperbolic Volume 4.05686 A-Polynomial See Data:10 132/A-polynomial

[edit Notes for 10 132's three dimensional invariants] 10 132 is a very interesting knot from the point of view of contact geometry. In particular, it is a transversely nonsimple knot, and it was the last knot with at most 10 crossings for which the maximal Thurston-Bennequin number was calculated.

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for 10 132's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-t+1-t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 0 }
Jones polynomial	$q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+3a^{4}$
Kauffman polynomial (db, data sources)	$a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-6a^{4}z^{6}-6a^{7}z^{5}-12a^{5}z^{5}-6a^{3}z^{5}+10a^{6}z^{4}+10a^{4}z^{4}+10a^{7}z^{3}+19a^{5}z^{3}+9a^{3}z^{3}-6a^{6}z^{2}-7a^{4}z^{2}-a^{2}z^{2}-5a^{7}z-8a^{5}z-4a^{3}z-az+2a^{6}+3a^{4}$
The A2 invariant	$-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$
The G2 invariant	$q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{86}-q^{84}-q^{82}-2q^{80}-q^{78}-q^{76}-2q^{74}-q^{68}-q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+2q^{52}+3q^{50}+q^{48}+q^{46}+2q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}-q^{28}+q^{26}-q^{24}-q^{18}+q^{16}-q^{12}+q^{4}-q^{2}+1+q^{-6}-q^{-8}$

Further Quantum Invariants

Further quantum knot invariants for 10_132.

A1 Invariants.

Weight	Invariant
1	$-q^{15}+q^{7}+q^{5}+q^{3}$
2	$q^{44}-q^{40}-q^{30}-q^{24}+q^{16}+q^{14}+q^{10}+q^{6}+2-q^{-4}$
3	$-q^{87}+q^{83}+q^{81}-q^{77}+q^{67}-q^{63}+q^{59}+q^{57}-q^{55}-q^{53}-q^{45}-q^{43}+q^{39}-q^{35}+q^{31}-q^{27}-q^{25}+q^{17}+2q^{15}+2q^{13}+q^{7}+2q^{5}+q^{3}-q-q^{-1}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$
1,1	$q^{60}+2q^{56}-2q^{50}-2q^{48}-2q^{44}+2q^{36}-2q^{34}-2q^{30}-q^{28}-2q^{24}+2q^{22}-q^{20}+4q^{18}+6q^{14}+q^{12}+4q^{10}+2q^{8}-2q^{6}+2q^{4}-2q^{2}+2-2q^{-2}$
2,0	$q^{58}+q^{56}+q^{54}-q^{48}-q^{46}-2q^{44}-2q^{42}-2q^{40}-q^{38}+q^{36}-q^{34}+q^{28}+q^{24}+2q^{22}+2q^{20}+q^{18}+q^{16}+q^{14}+q^{10}+q^{8}+q^{4}+q^{2}+1-q^{-2}-q^{-4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{46}+q^{42}+q^{40}-2q^{36}-2q^{34}-4q^{32}-5q^{30}-2q^{28}-q^{26}+2q^{24}+3q^{22}+6q^{20}+4q^{18}+3q^{16}+2q^{14}+q^{12}-q^{10}-q^{8}-q^{4}+1$
1,0,0	$-q^{29}-q^{27}-2q^{25}-q^{23}+q^{19}+2q^{17}+2q^{15}+2q^{13}+q^{11}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{60}+q^{58}+2q^{56}+2q^{54}+2q^{52}+q^{50}-q^{48}-4q^{46}-6q^{44}-8q^{42}-9q^{40}-8q^{38}-5q^{36}+4q^{32}+8q^{30}+11q^{28}+11q^{26}+8q^{24}+6q^{22}+q^{20}-q^{18}-3q^{16}-2q^{14}-2q^{12}-q^{10}+q^{6}+1$
1,0,0,0	$-q^{36}-q^{34}-2q^{32}-2q^{30}-q^{28}+q^{24}+2q^{22}+3q^{20}+2q^{18}+2q^{16}+q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{46}-q^{42}-q^{40}+q^{30}+q^{26}+q^{22}+q^{16}+q^{12}+q^{10}+q^{8}+q^{4}-1$
1,0	$q^{76}+q^{68}-q^{58}-2q^{56}-q^{54}-q^{52}-2q^{50}-2q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2q^{34}+2q^{32}+2q^{30}+q^{28}+2q^{26}+2q^{24}+q^{22}+q^{18}-q^{12}-q^{4}+1$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{62}+q^{58}+q^{56}+q^{54}-q^{50}-2q^{48}-4q^{46}-4q^{44}-5q^{42}-4q^{40}-3q^{38}+q^{34}+4q^{32}+5q^{30}+6q^{28}+5q^{26}+4q^{24}+3q^{22}+q^{20}+q^{18}-q^{16}-q^{14}-q^{12}-q^{8}+1$

G2 Invariants.

Weight	Invariant
1,0	$q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{86}-q^{84}-q^{82}-2q^{80}-q^{78}-q^{76}-2q^{74}-q^{68}-q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+2q^{52}+3q^{50}+q^{48}+q^{46}+2q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}-q^{28}+q^{26}-q^{24}-q^{18}+q^{16}-q^{12}+q^{4}-q^{2}+1+q^{-6}-q^{-8}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 132"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{2}-t+1-t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}+3z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 5, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+3a^{4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-6a^{4}z^{6}-6a^{7}z^{5}-12a^{5}z^{5}-6a^{3}z^{5}+10a^{6}z^{4}+10a^{4}z^{4}+10a^{7}z^{3}+19a^{5}z^{3}+9a^{3}z^{3}-6a^{6}z^{2}-7a^{4}z^{2}-a^{2}z^{2}-5a^{7}z-8a^{5}z-4a^{3}z-az+2a^{6}+3a^{4}$

Vassiliev invariants

V2 and V3:

(3, -5)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $12$ $-40$ $72$ $174$ $26$ $-480$ $-{\frac {2608}{3}}$ $-{\frac {352}{3}}$ $-168$ $288$ $800$ $2088$ $312$ ${\frac {42751}{10}}$ $-{\frac {3506}{15}}$ ${\frac {10034}{5}}$ ${\frac {203}{2}}$ ${\frac {2751}{10}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 10 132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 χ -1             1 1 0 -3               1 1 -5         1 2     1 -7       1         1 -9       1 1       0 -11   1 1           0 -13                 0 -15 1               -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 132]]
Out[2]=	10
In[3]:=	PD[Knot[10, 132]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[15, 18, 16, 19], X[9, 16, 10, 17], X[17, 10, 18, 11], X[13, 20, 14, 1], X[19, 14, 20, 15], X[11, 6, 12, 7], X[2, 8, 3, 7]]
In[4]:=	GaussCode[Knot[10, 132]]
Out[4]=	GaussCode[1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7]
In[5]:=	BR[Knot[10, 132]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, -2, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 132]][t]
Out[6]=	-2 1 2 1 + t - - - t + t t
In[7]:=	Conway[Knot[10, 132]][z]
Out[7]=	2 4 1 + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[5, 1], Knot[10, 132]}
In[9]:=	{KnotDet[Knot[10, 132]], KnotSignature[Knot[10, 132]]}
Out[9]=	{5, 0}
In[10]:=	J=Jones[Knot[10, 132]][q]
Out[10]=	-7 -6 -5 -4 -2 -q + q - q + q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[5, 1], Knot[10, 132]}
In[12]:=	A2Invariant[Knot[10, 132]][q]
Out[12]=	-22 -20 -18 -14 -12 2 -8 -6 -q - q - q + q + q + --- + q + q 10 q
In[13]:=	Kauffman[Knot[10, 132]][a, z]
Out[13]=	4 6 3 5 7 2 2 4 2 3 a + 2 a - a z - 4 a z - 8 a z - 5 a z - a z - 7 a z - 6 2 3 3 5 3 7 3 4 4 6 4 6 a z + 9 a z + 19 a z + 10 a z + 10 a z + 10 a z - 3 5 5 5 7 5 4 6 6 6 3 7 5 7 6 a z - 12 a z - 6 a z - 6 a z - 6 a z + a z + 2 a z + 7 7 4 8 6 8 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 132]], Vassiliev[3][Knot[10, 132]]}
Out[14]=	{0, -5}
In[15]:=	Kh[Knot[10, 132]][q, t]
Out[15]=	-3 1 1 1 1 1 1 1 1 q + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 11 6 11 5 9 4 7 4 9 3 5 3 q t q t q t q t q t q t q t 2 1 ----- + --- 5 2 q t q t