9 47

9_46

9_48

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9 47 Quick Notes

Simple square depiction

Threefold symmetrical depiction

Ornate threefold symmetrical depiction

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X11,1,12,18 X17,13,18,12
Gauss code	1, -4, 3, -7, 6, -1, 2, -3, 5, -6, -8, 9, 7, -5, 4, -2, -9, 8
Dowker-Thistlethwaite code	6 8 10 16 14 -18 4 2 -12
Conway Notation	[8*-20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-2][-7] Hyperbolic Volume 10.05 A-Polynomial See Data:9 47/A-polynomial

[edit Notes for 9 47's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for 9 47's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+6t-5+6t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 27, 2 }
Jones polynomial	$2q^{5}-4q^{4}+4q^{3}-5q^{2}+5q-3+3q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+4z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$
Kauffman polynomial (db, data sources)	$2z^{7}a^{-1}+2z^{7}a^{-3}+6z^{6}a^{-2}+3z^{6}a^{-4}+3z^{6}+az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}-16z^{4}a^{-2}-7z^{4}a^{-4}-9z^{4}-2az^{3}+z^{3}a^{-1}+6z^{3}a^{-3}+3z^{3}a^{-5}+11z^{2}a^{-2}+9z^{2}a^{-4}+3z^{2}a^{-6}+5z^{2}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$
The A2 invariant	$-q^{6}+q^{4}+q^{2}+2+2q^{-2}-q^{-4}+q^{-6}-2q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
The G2 invariant	$q^{32}-2q^{30}+4q^{28}-7q^{26}+4q^{24}-q^{22}-8q^{20}+16q^{18}-16q^{16}+16q^{14}-4q^{12}-11q^{10}+20q^{8}-18q^{6}+14q^{4}-2q^{2}-13+21q^{-2}-8q^{-4}+13q^{-8}-19q^{-10}+18q^{-12}-4q^{-14}-9q^{-16}+12q^{-18}-21q^{-20}+25q^{-22}-13q^{-24}+9q^{-28}-19q^{-30}+23q^{-32}-18q^{-34}+4q^{-36}+3q^{-38}-14q^{-40}+19q^{-42}-11q^{-44}-3q^{-46}+15q^{-48}-19q^{-50}+12q^{-52}+q^{-54}-17q^{-56}+20q^{-58}-17q^{-60}+11q^{-62}+2q^{-64}-11q^{-66}+14q^{-68}-12q^{-70}+8q^{-72}-5q^{-76}+2q^{-78}-2q^{-80}+2q^{-82}-q^{-84}+2q^{-86}+q^{-88}$

Further Quantum Invariants

Further quantum knot invariants for 9_47.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+2q^{3}+2q^{-1}-q^{-5}-2q^{-9}+2q^{-11}$
2	$q^{16}-2q^{14}-3q^{12}+5q^{10}+2q^{8}-5q^{6}+3q^{4}+6q^{2}-4-q^{-2}+4q^{-4}-3q^{-6}-4q^{-8}+2q^{-10}+2q^{-12}-3q^{-14}-q^{-16}+7q^{-18}-q^{-20}-5q^{-22}+6q^{-24}-5q^{-28}+q^{-30}+q^{-32}$
3	$-q^{33}+2q^{31}+3q^{29}-2q^{27}-8q^{25}-5q^{23}+11q^{21}+11q^{19}-5q^{17}-17q^{15}-2q^{13}+20q^{11}+15q^{9}-14q^{7}-20q^{5}+6q^{3}+24q-q^{-1}-24q^{-3}-7q^{-5}+19q^{-7}+10q^{-9}-18q^{-11}-8q^{-13}+13q^{-15}+12q^{-17}-9q^{-19}-10q^{-21}+4q^{-23}+16q^{-25}+3q^{-27}-16q^{-29}-13q^{-31}+13q^{-33}+21q^{-35}-10q^{-37}-27q^{-39}+q^{-41}+27q^{-43}+5q^{-45}-17q^{-47}-12q^{-49}+11q^{-51}+10q^{-53}-3q^{-55}-4q^{-57}-2q^{-59}+2q^{-61}$
4	$q^{56}-2q^{54}-3q^{52}+2q^{50}+5q^{48}+11q^{46}-2q^{44}-16q^{42}-18q^{40}-10q^{38}+29q^{36}+33q^{34}+10q^{32}-26q^{30}-58q^{28}-16q^{26}+35q^{24}+70q^{22}+43q^{20}-51q^{18}-83q^{16}-45q^{14}+56q^{12}+109q^{10}+32q^{8}-71q^{6}-116q^{4}-24q^{2}+98+97q^{-2}-5q^{-4}-109q^{-6}-72q^{-8}+49q^{-10}+97q^{-12}+32q^{-14}-69q^{-16}-68q^{-18}+17q^{-20}+70q^{-22}+30q^{-24}-42q^{-26}-53q^{-28}+54q^{-32}+32q^{-34}-20q^{-36}-52q^{-38}-34q^{-40}+32q^{-42}+60q^{-44}+38q^{-46}-47q^{-48}-95q^{-50}-23q^{-52}+75q^{-54}+116q^{-56}+9q^{-58}-120q^{-60}-102q^{-62}+18q^{-64}+135q^{-66}+84q^{-68}-58q^{-70}-108q^{-72}-55q^{-74}+60q^{-76}+84q^{-78}+16q^{-80}-38q^{-82}-49q^{-84}-6q^{-86}+22q^{-88}+17q^{-90}+4q^{-92}-8q^{-94}-5q^{-96}+q^{-98}+q^{-100}$
5	$-q^{85}+2q^{83}+3q^{81}-2q^{79}-5q^{77}-8q^{75}-4q^{73}+7q^{71}+24q^{69}+24q^{67}+3q^{65}-27q^{63}-51q^{61}-48q^{59}-5q^{57}+66q^{55}+95q^{53}+67q^{51}-13q^{49}-112q^{47}-156q^{45}-90q^{43}+58q^{41}+190q^{39}+220q^{37}+93q^{35}-137q^{33}-303q^{31}-264q^{29}-17q^{27}+278q^{25}+409q^{23}+230q^{21}-146q^{19}-452q^{17}-434q^{15}-65q^{13}+380q^{11}+548q^{9}+284q^{7}-221q^{5}-572q^{3}-451q+40q^{-1}+495q^{-3}+539q^{-5}+130q^{-7}-371q^{-9}-539q^{-11}-236q^{-13}+254q^{-15}+480q^{-17}+282q^{-19}-146q^{-21}-402q^{-23}-276q^{-25}+86q^{-27}+317q^{-29}+229q^{-31}-48q^{-33}-254q^{-35}-192q^{-37}+48q^{-39}+206q^{-41}+151q^{-43}-34q^{-45}-183q^{-47}-159q^{-49}+13q^{-51}+173q^{-53}+191q^{-55}+55q^{-57}-150q^{-59}-255q^{-61}-160q^{-63}+92q^{-65}+319q^{-67}+312q^{-69}+19q^{-71}-345q^{-73}-469q^{-75}-191q^{-77}+296q^{-79}+590q^{-81}+398q^{-83}-168q^{-85}-624q^{-87}-572q^{-89}-30q^{-91}+534q^{-93}+663q^{-95}+242q^{-97}-364q^{-99}-619q^{-101}-370q^{-103}+129q^{-105}+472q^{-107}+404q^{-109}+42q^{-111}-276q^{-113}-319q^{-115}-135q^{-117}+99q^{-119}+199q^{-121}+136q^{-123}+q^{-125}-89q^{-127}-78q^{-129}-31q^{-131}+18q^{-133}+35q^{-135}+18q^{-137}-3q^{-139}-6q^{-141}-4q^{-143}-2q^{-145}+2q^{-147}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+q^{4}+q^{2}+2+2q^{-2}-q^{-4}+q^{-6}-2q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
1,1	$q^{20}-4q^{18}+10q^{16}-22q^{14}+32q^{12}-46q^{10}+56q^{8}-52q^{6}+48q^{4}-22q^{2}+4+36q^{-2}-58q^{-4}+74q^{-6}-94q^{-8}+90q^{-10}-94q^{-12}+70q^{-14}-52q^{-16}+28q^{-18}+5q^{-20}-24q^{-22}+50q^{-24}-56q^{-26}+58q^{-28}-46q^{-30}+32q^{-32}-22q^{-34}+10q^{-36}-2q^{-38}-4q^{-40}+2q^{-46}$
2,0	$q^{18}-q^{16}-3q^{14}-q^{12}+2q^{10}+2q^{8}+4q^{4}+6q^{2}+3-2q^{-2}+q^{-4}-3q^{-6}-3q^{-8}-2q^{-10}-2q^{-12}-2q^{-14}-2q^{-16}+3q^{-18}+3q^{-24}+5q^{-26}-q^{-28}-q^{-30}+q^{-32}-2q^{-38}-q^{-48}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-2q^{12}+q^{8}-4q^{6}+5q^{4}+3q^{2}+1+7q^{-2}+3q^{-4}-4q^{-6}-2q^{-8}-3q^{-10}-5q^{-12}-2q^{-14}+5q^{-18}+q^{-20}+q^{-22}+5q^{-24}-4q^{-26}-3q^{-28}+3q^{-30}-3q^{-32}-q^{-34}+3q^{-36}$
1,0,0	$-q^{7}+q^{5}+3q+q^{-1}+2q^{-3}-q^{-11}-2q^{-15}-2q^{-19}+q^{-21}+q^{-25}+q^{-27}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}-q^{14}-q^{12}-2q^{8}+4q^{4}+3q^{2}+3+9q^{-2}+7q^{-4}+q^{-6}-2q^{-8}+q^{-10}-6q^{-12}-12q^{-14}-5q^{-16}-6q^{-20}+q^{-22}+9q^{-24}+2q^{-26}+2q^{-28}+6q^{-30}+3q^{-32}-4q^{-34}-q^{-36}+2q^{-38}-3q^{-40}-5q^{-42}+q^{-44}+2q^{-46}-2q^{-48}+q^{-50}+2q^{-52}$
1,0,0,0	$-q^{8}+q^{6}+2q^{2}+2+q^{-2}+2q^{-4}+q^{-8}-q^{-10}+q^{-12}-q^{-14}-2q^{-18}-q^{-20}-q^{-22}-2q^{-24}+q^{-26}+q^{-30}+q^{-32}+q^{-34}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+2q^{12}-4q^{10}+5q^{8}-4q^{6}+5q^{4}-3q^{2}+3+q^{-2}-q^{-4}+6q^{-6}-6q^{-8}+9q^{-10}-9q^{-12}+8q^{-14}-8q^{-16}+3q^{-18}-3q^{-20}-q^{-22}+q^{-24}-4q^{-26}+5q^{-28}-5q^{-30}+5q^{-32}-3q^{-34}+3q^{-36}$
1,0	$q^{24}-2q^{20}-2q^{18}+2q^{16}+3q^{14}-3q^{12}-4q^{10}+q^{8}+7q^{6}+3q^{4}-3q^{2}-1+6q^{-2}+5q^{-4}+q^{-6}-4q^{-8}-q^{-10}+q^{-12}-q^{-14}-6q^{-16}-4q^{-18}+q^{-20}+q^{-22}-2q^{-24}-4q^{-26}+3q^{-28}+6q^{-30}+2q^{-32}-3q^{-34}+q^{-36}+5q^{-38}+3q^{-40}-4q^{-42}-5q^{-44}+q^{-46}+4q^{-48}-q^{-50}-4q^{-52}-2q^{-54}+q^{-56}+3q^{-58}$

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-2q^{30}+4q^{28}-7q^{26}+4q^{24}-q^{22}-8q^{20}+16q^{18}-16q^{16}+16q^{14}-4q^{12}-11q^{10}+20q^{8}-18q^{6}+14q^{4}-2q^{2}-13+21q^{-2}-8q^{-4}+13q^{-8}-19q^{-10}+18q^{-12}-4q^{-14}-9q^{-16}+12q^{-18}-21q^{-20}+25q^{-22}-13q^{-24}+9q^{-28}-19q^{-30}+23q^{-32}-18q^{-34}+4q^{-36}+3q^{-38}-14q^{-40}+19q^{-42}-11q^{-44}-3q^{-46}+15q^{-48}-19q^{-50}+12q^{-52}+q^{-54}-17q^{-56}+20q^{-58}-17q^{-60}+11q^{-62}+2q^{-64}-11q^{-66}+14q^{-68}-12q^{-70}+8q^{-72}-5q^{-76}+2q^{-78}-2q^{-80}+2q^{-82}-q^{-84}+2q^{-86}+q^{-88}$

.

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["9 47"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-4t^{2}+6t-5+6t^{-1}-4t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+2z^{4}-z^{2}+1$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$2q^{5}-4q^{4}+4q^{3}-5q^{2}+5q-3+3q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+4z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{7}a^{-1}+2z^{7}a^{-3}+6z^{6}a^{-2}+3z^{6}a^{-4}+3z^{6}+az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}-16z^{4}a^{-2}-7z^{4}a^{-4}-9z^{4}-2az^{3}+z^{3}a^{-1}+6z^{3}a^{-3}+3z^{3}a^{-5}+11z^{2}a^{-2}+9z^{2}a^{-4}+3z^{2}a^{-6}+5z^{2}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$

Vassiliev invariants

V2 and V3:

(-1, -2)

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 $-4$ $-16$ $8$ $-{\frac {110}{3}}$ $-{\frac {34}{3}}$ $64$ ${\frac {32}{3}}$ ${\frac {32}{3}}$ $-16$ $-{\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ ${\frac {136}{3}}$ ${\frac {10529}{30}}$ $-{\frac {418}{15}}$ ${\frac {10738}{45}}$ $-{\frac {833}{18}}$ ${\frac {1409}{30}}$

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=$2 is the signature of 9 47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

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[9, 47]]
Out[2]=	9
In[3]:=	PD[Knot[9, 47]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[8, 3, 9, 4], X[2, 15, 3, 16], X[14, 9, 15, 10], X[10, 6, 11, 5], X[4, 14, 5, 13], X[11, 1, 12, 18], X[17, 13, 18, 12]]
In[4]:=	GaussCode[Knot[9, 47]]
Out[4]=	GaussCode[1, -4, 3, -7, 6, -1, 2, -3, 5, -6, -8, 9, 7, -5, 4, -2, -9, 8]
In[5]:=	BR[Knot[9, 47]]
Out[5]=	BR[4, {-1, 2, -1, 2, 3, 2, -1, 2, 3}]
In[6]:=	alex = Alexander[Knot[9, 47]][t]
Out[6]=	-3 4 6 2 3 -5 + t - -- + - + 6 t - 4 t + t 2 t t
In[7]:=	Conway[Knot[9, 47]][z]
Out[7]=	2 4 6 1 - z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 47]}
In[9]:=	{KnotDet[Knot[9, 47]], KnotSignature[Knot[9, 47]]}
Out[9]=	{27, 2}
In[10]:=	J=Jones[Knot[9, 47]][q]
Out[10]=	-2 3 2 3 4 5 -3 - q + - + 5 q - 5 q + 4 q - 4 q + 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 47]}
In[12]:=	A2Invariant[Knot[9, 47]][q]
Out[12]=	-6 -4 -2 2 4 6 8 12 14 16 20 2 - q + q + q + 2 q - q + q - 2 q - q - q + q + q
In[13]:=	Kauffman[Knot[9, 47]][a, z]
Out[13]=	2 2 2 -6 2 -2 3 z 5 z 2 z 2 3 z 9 z 11 z 1 - a - -- - a - --- - --- - --- + 5 z + ---- + ---- + ----- + 4 5 3 a 6 4 2 a a a a a a 3 3 3 4 4 5 5 5 3 z 6 z z 3 4 7 z 16 z z 4 z 4 z ---- + ---- + -- - 2 a z - 9 z - ---- - ----- + -- - ---- - ---- + 5 3 a 4 2 5 3 a a a a a a a 6 6 7 7 5 6 3 z 6 z 2 z 2 z a z + 3 z + ---- + ---- + ---- + ---- 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[9, 47]], Vassiliev[3][Knot[9, 47]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 47]][q, t]
Out[15]=	3 1 2 1 1 2 q 3 5 4 q + 2 q + ----- + ----- + ---- + --- + --- + 2 q t + 3 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 11 4 2 q t + 2 q t + 2 q t + 2 q t + 2 q t